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TREATISE 



ox 



ASTRONOMY 



DESCRIPTIVE, PHYSICAL, AND PRACTICAL. 



DESIGNED FOR 



SCHOOLS, COLLEGES, AND PRIVATE STUDENTS. 




BY H. N. ROBINSON, A.M., 

FORMERLY PROFESSOR OF MATHEMATICS IN THE U. S. NAVY ; AUTHOR OF A TREATISE 
OH ARITHMETIC; ALGEBRA; NATURAL PHILOSOPHY ; ETC. 



ALBANY, NEW YOKK: 

ERASTUS H. PEASE & CO., 82 STATE STREET, 

CINCINNATI: 
JACOB ERNST, NO. 183 MAIN STREET. 



,1849 






Entered, according to act of Congress, in the year 1849, 

By HORATIO N. ROBINSON, 

In the Clerk's Office of the District Court of the United States, for the 
District of Ohio. 



PREFACE. 



To give at once a clear explanation of the design and in- p REFACE . 

tended character of this work, it is import-ant to state that its 

author, in early life, imbibed quite a passion for astronomy, 
and, of course, he naturally sought the aid of books ; but, in 
this field of research, he was really astonished to find how 
little substantial aid he could procure from that source, and 
not even to this day have his desires been gratified. 

Then, as now, books of great worth and high merit were to 
be found, but they did not meet the wants of a learner ; the 
substantially good were too voluminous and mathematically 
abstruse to be much used by the humble pupil, and the less 
mathematical were too superficial and trifling to give satis- 
faction to the real aspirant after astronomical knowledge. 

Of the less mathematical and more elaborate works on as- 
tronomy there are two classes — the pure and valuable, like 
the writings of Biot and Herschel; but, excellent as these 
are, they are not adapted to the purposes of instruction; and 
every effort to make class books of them has substantially 
failed. From the other class, which consists of essays and 
popular lectures, little substantial knowledge can be gathered, 
for they do not teach astronomy ; as a general thing, they only 
glorify it; they may excite our wonder concerning the im- 
mensity or grandeur of the heavens, but they give us no ad- 
ditional power to investigate the science. 

Another class of more brief and valuable productions were, 
and are always to be found, in which most of the important 
facts are recorded; such as the distances, magnitudes, and mo- 
tions of the heavenly bodies; but how these facts became 
known is rarely explained : this is what the true searcher after 
science will always demand, and this book is designed ex- 
pressly to meet that demand. 

In the first part of the book we suppose the reader entirely 
unacquainted with the subject ; but we suppose him compe- 
tent to the task — to be, at least, sixteen years of age — to have 
a good knowledge of proportion, some knowledge of algebra, 
geometry, and trigonometry — and then, and not until then, 
can the study be pursued with any degree of success worth 
mentioning. Such a person, and with such acquirements as 

( iii ) 



Iv PREFACE. 

Preface, we have here designated, we believe, can take this book and 
learn astronomy in comparatively a short time; for the chief 
design of the work is, to teach whoever desires to learn : and 
it matters not where the learner may be, in a college, 
academy, school, or a solitary student at home, and alone in 
the pursuit. 

The book is designed for two classes of students — the well 
prepared in the mathematics, and the less prepared ; the for- 
mer are expected to read the text notes, the latter should 
omit them. With the text notes, we conceive it, or rather 
designed it to be, a very suitable book to give sound elemen- 
tary instruction in astronomy ; but we do not offer the work 
as complete on practical astronomy; for whoever becomes a 
practical astronomer will, of course, seek the aid of complete 
and elaborate sets of tables, such as would be improper to 
insert in a school book. 

We have inserted tables only for the purpose of carrying 
out a sound theoretical plan of instruction, and, therefore, we 
have given as few as possible, and those few in a very con- 
tracted form. The epochs for the sun and moon may be ex- 
tended forward or backward, to any extent, by any one who 
understands the theory. 

The chapters on comets, variable stars, &c, are compila- 
tions, and are printed in smaller type; and the works to 
which we are most indebted, are Herschel's Astronomy and 
the Cambridge Astronomy, originally the work of M. Biot. 

Other parts of the work, we believe, will be admitted as 
mainly original, by all who take pains to examine it. 

The chief merits claimed for this book are, brevity, clear- 
ness of illustration, anticipating the difficulties of the pupil, 
and removing them, and bringing out all the essential points 
of the scienoe. 

Some originality is claimed, also, in several of our illustra- 
tions, particularly that of showing the rationale of tides rising 
on the opposite sides of the earth from the moon ; and in the 
general treatment of eclipses ; but it is for others to deter- 
mine how much merit should be awarded for such originali- 
ties; we have, however, used greater conciseness and per- 
spicuity in general computations than is to be found in most 
of the books on this subject ; and this last remark will apply 
to the whole work. 



CONTENTS. 



SECTION I. 

Page. 

Introduction. — Definition of terms, &c, 1 Contents 

CHAPTER I. 

Preliminary Observations, 6 

A fixed point in the heavens — the pole and polar star, 7 

Index to the length of one year, 11 

Fixed stars — why so called, 12 

CHAPTER II. 

Appearances in the heavens, 13 

Important instruments for an Observatory, 14 

Standard measure for time, 16 

An astronomical clock, 17 

Movable and wandering bodies, 19 

To find the right ascension of the sun, moon, and planets, 21 

CHAPTER III. 

Refraction — position of the equinox, &c, 21 

Altitude and azimuth instrument, 23 

Astronomical refraction — its effect, &c, 25 

The declination of a star — how found, 29 

Observations to find the equinox, 33 

Length of the year— how observed, 36 

CHAPTER IV. 

Geography of the heavens, 38 

Method of tracing the stars, 41 

What constitutes a definite description, 43 

How to find the right ascension of any star, 44 

The southern cross and Magellan clouds, 48 



SECTION II. 
DESCRIPTIVE ASTRONOMY 

CHAPTER I. 
First consideration of the distances to the heavenly bodies — size 

and figure of the earth, 49 

How to find the diameter of the earth, 53 

(v) 



vi CONTENTS. 

Contents. Page. 

Dip of the horizon, 54 

The exact dimensions of the earth, 56 

Gravity of the earth diminished on its surface by its rotation,. . . 58 

A degree between two meridians — the law of decrease, 61 

CHAPTER II. 

Parallax, general and horizontal, 62 

Relation between parallax and distance, 63 

Lunar parallax — how found, 65 

Variable distance to the moon, 67 

Apogee and perigee, 67 

Mean parallax — and parallax at mean distance, 68 

Mean distance to the moon, 69 

Connection between semidiameter and the horizontal parallax of 

any celestial body, 70 

The earth a moon to the moon, 71 

CHAPTER III. 

The earth's orbit eccentric, &c, 72 

Methods of measuring apparent diameters, 73 

Eccentricity of the earth's orbit — how found, 76 

Variations in solar motion, 76 

Eccentricity of orbit and greatest equation of center connected, . . 85 

CHAPTER IV. 

Causes of the change of seasons, 87 

Temperature of the earth, 89 

Times of extreme temperature, 90 

CHAPTER V. 

Equation of time, 90 

Mean and apparent noon, 91 

What is meant by sun slow and sun fast, 94 

Use of the equation of time, 96 

CHAPTER VI. 

Apparent motions of the planets, 97 

The morning and evening star, 98 

Motion of Venus in respect to the fixed stars, 101 

Retrograde motion of planets accounted for, 102 

CHAPTER VII. 
First approximations of the relative distances of the planets from 

the sun, 104 

What to understand by stationary, 106 

Method of approximating to the orbits of planets-, 108 



CONTENTS. vii 

CHAPTER VIII. 

Page. 

Methods of observing the periodical revolutions of the planets,. . Ill Contents 

Diurnal motion of the planets, 114 

Times of revolution and distances compared, 117 

Kepler's Laws, 118 

CHAPTER IX. 

Transits of Venus and Mercury, 119 

Periods of the transits of Venus, 120, 121, 122 

Deductions from a transit made plain, 124 

CHAPTER X. 

The horizontal parallaxes of the planets computed, 127 

Real distance between the earth and sun determined, 128 

How to find the magnitudes of the planets, 130 

CHAPTER XI. 

A general description of the planets, 131 

Professor Bode's law of planetary distances, 134 

A bold hypothesis, 135 

Progressive nature of light — how determined, 139 

CHAPTER XII. 

Comets, 144 

Inclinations of their orbits, 147 

Fears anciently entertained concerning comets, 150 

CHAPTER XIII 
On the peculiarities of the fixed stars, 150 



SECTION III. 

PHYSICAL ASTRONOMY. 

CHAPTER I. 

General laws of motion — theory of gravity, 157 

Attraction of a sphere — of a spherical shell, &c, 161 

A general expression for the mutual attraction of two bodies,. . . . 162 

CHAPTER II. 

Demonstration of Kepler's Laws, 163 

A common error, 167 

How a planet finds its own orbit, 168 

Kepler's third law rigorously true in circles and ellipses, 171 



viii CONTENTS. 

CHAPTER III. 

Page. 

Contents. Masses of the planets, &c, s 174 

The diameter of the earth accurately determined from equations 

in physical astronomy, ( Art. 171 ), 180 

The mass of the moon determined — densities of bodies, 183 

CHAPTER IV. 

Lunar Perturbations, 185 

Cause of nutation, 187 

Mean radial force, 191 

Acceleration of the moon's mean motion, 197 

A summary statement of the cause, 198 

The true mean value of the radial force, 199 

A summary statement of the lunar irregularities, 200 

CHAPTER V. 

The tides, 201 

A summary illustration of the physical cause of tides, 202 

Mass of the moon computed from the tides, 204 

CHAPTER VI. 

Planetary perturbations, 206 

Action and reaction equal and contrary, 206 

The effects of commensurate revolutions of the planets, 208 

The great inequalities of Jupiter and Saturn, 209 

These inequalities explained, 209, 210, 211, 212 

The physical effects on Uranus that led to the discovery of Neptune, 213 

CHAPTER VII. 

Aberration — nutation, and precession of the equinoxes, 213 

The velocity of light computed from the effect of aberration,. . . . 215 

Cause of nutation explained, 218 

The physical cause of the precession of the equinoxes, 222 

Proper motion of the stars — how found, 224 

The latitude of the sun explained, 226 



SECTION IV. 
PRACTICAL ASTRONOMY. 

Preparatory remarks and trigonometrical formula?, 229 

CHAPTER I. 

Problems in relation to the sphere, 232 

To find the time, from the latitude of the place — altitude, and 
declination of the sun, 239 



CONTENTS. 



IX 



Page. 

An artificial horizon, 242 Contexts. 

Absolute and local time, 244 

Lunar observations, 246 

Proportional logarithms, 247 

CHAPTER II. 

Explanation of the tables, 250 

To compute the sun's longitude, 253 

To find the equation of time to great exactness, 254 

To compute the time of new and full moons, 255 

Eclipses — when they occur, &c, 257 

Limits of eclipses, 259 

Periods of eclipses, 261, 263 

Elements of lunar eclipses, 264 

Semidiameter of the earth's shadow, 265 

CHAPTER III. 

Preparation for the computation of eclipses, 267 

Directions for computing the moon's longitude, latitude, &c.,. . . 268 

To construct a lunar eclipse, 275 

To make exact computations in respect to lunar eclipses, 277 

CHAPTER IV. 

Solar eclipses — general and local, 279 

Elements for the computation of a solar eclipse, 280 

To construct a general eclipse, 281 

How to determine the duration of a solar eclipse on the earth, . . . 282 

To find where the sun will be centrally eclipsed at noon, 286 

Results taken from the projection, 287 

Results from trigonometrical computations, 288 

CHAPTER V. 

Local eclipses, &c, 291 

How to construct a local solar eclipse, 291 

How to find the time of greatest obscuration, 295 

To find the time of the beginning, end, &c, of a local eclipse by 

the application of analytical geometry, 296 

Catalogue of eclipses which will take place between the years 

1850 and 1900, 300 



ASTEONOMY. 



INTRODUCTION. 



Astronomy is the science which treats of the heavenly Astronomy 
bodies, describes their appearances, determines their magni- e ne 
tndes, and discovers the laws which govern their motions. 

When we merely state facts and describe appearances as The dw 
they exist in the heavens, we call it Descriptive Astronomy. S10ns ot as " 

. . tronomy. 

When we compnte magnitudes, determine distances, record 
observations, and make any computations whatever, we call 
it Practical Astronomy. 

The investigation of the laws which govern the celestial 
motions, and the explanation of the causes which bring about 
the known results, is called Physical Astronomy. 

When the mariner makes use of the index of the heavens, Nautical 
to determine his position on the earth, such observations, and as ronomy - 
their corresponding computations, are called Nautical Astro- 
nomy. 

By nautical astronomy we determine positions on the Geography 
earth, and subsequently, the magnitude of the earth : and an a * tr ° n0 " 

i -v o * m y united. 

thus, we perceive, that Geography and Astronomy must be 
linked together ; and no one c.an fully understand the former 
science, without the aid of the latter. 

Astronomy is the most ancient of all the sciences, for, in T '» e anti 
the earliest age, the people could not have avoided observing ^0° 
the successive returns of day and night, and summer and 
winter. They could not fail to perceive that short days cor- 
responded to winter, and long days to summer ; and it was 
thus, probably, that the attentions of men were first drawn 
to the study of astronomy. 

(1) 



2 ASTRONOMY. 

Introduc. In this work, we shall not take facts unless they are within 

Facts alone the sphere of our own observations. We shall not perempto- 

not science. T ^ gtate ^ the eart k ig 79-^2 miles in diameter; that the 

moon is about 240,000 miles from the earth, and the sun 
95,000,000 of miles; for such facts, alone, and of themselves, do 
not constitute knowledge, though often mistaken for knowledge. 
We shall direct the mind of the reader, step by step, through 
the observations and through the investigations, so that he 
can decide for himself that the earth must be of such a mag- 
nitude, and is thus far from the other heavenly bodies ; and 
that will be knowledge of the most essential kind. 
The foun- A\\ astronomical knowledge has its foundation in observa- 
astronomicai ^ on ' an( ^ * ne nrs ^ 0D J ec t of this book shall be to point out 
knowledge, what observations must be taken, and what deductions must 
be made therefrom ; but the great book which the pupil must 
study, if he would meet with success, is the one which spreads 
out its pages on the blue arch above ; and he must place but 
secondary dependence on any book that is merely the work 
of human art. 

As we disapprove of the practice of throwing to the reader 
astounding astronomical facts, whether he can digest them or 
not, and as we are to take the inductive method, and to lead 
the student by the hand, we must commence on the supposi- 
tion that the reader is entirely unacquainted even with the 
common astronomical facts, and now for the first time seriously 
brings his mind to the study of the subject ; but we shall 
suppose some maturity of mind, and some preparation, by the 
acquisition of at least respectable mathematical knowledge. 
Conven- Every science has its technicalities and conventional terms ; 

tional terms , . . . , -, 

and defini- anc * astronomy is by no means an exception to the general 
tions. rule ; and as it will prepare the way for a clearer understand- 

ing of our subject, we now give a short list of some of the 
technical terms, which must be used in our composition. 

Horizon. — Every person, wherever he may be, conceives 
himself to be in the center of a circle ; and the circumference 
of that circle is where the earth and sky apparently meet. 
That circle is called the horizon, 



INTRODUCTION. 3 

Altitude. — The perpendicular hight from the horizon, intboduc. 
measured by degrees of a circle. 

Meridian. — An imaginary line, north and south from any 
point or place, whether it is conceived to run along the earth 
or through the heavens. If the meridian is conceived to 
divide "both the earth and the heavens, it is then considered 
as a plane, and is spoken of as the plane of the meridian. 

Poles. — The points where all meridians come together : 
poles of the earth — the extremities of the earth's axis. 

Zenith. — The zenith of any place, is the point directly Poles of 
overhead ; and the Nadir is directly opposite to the zenith, or l e honzon * 
under our feet. The zenith and nadir are the poles to the 
horizon. 

Verticals. — All lines passing from the zenith, perpendicu- Prime ver. 
lar to the horizon, are called Verticals, or Vertical Circles. tlca1. 
The one passing at right angles to the meridian, and striking 
the horizon at the east and west points, is called the Prime 
Vertical. 

Azimuth. — The angular position of a body from the meri- 
dian, measured on the circle of the horizon, is called its Azi- 
muth. 

The angular position, measured from its prime vertical, is Amplitude. 
called its Amplitude. 

The sum of the azimuth and amplitude must always make 
90 degrees. 

Equator. — The Earth's Equator is a great circle, east and 
west, and equidistant from the poles, dividing the earth into 
two hemispheres, a northern, and a southern. 

The Celestial Equator is the plane of the earth's equator celestial 
conceived to extend into the heavens. equator. 

When the sun, or any other heavenly body, meets the Equinoc 
celestial equator, it is said to be in the Equinox, and the tia1 ' 
equatorial line in the heavens is called the Equinoctial. 

Latitude. — The latitude of any place on the earth, is 
its distance from the equator, measured in degrees on the 
meridian, either north or south. 

If the measure is toward the north, it is north latitude; if 
toward the couth, south latitude. 



4 ASTRONOMY. 

Lntroduc . The distance from the equator to the poles is 90 degrees — 
one-fourth of a circle ; and we shall know the circumference 
of the whole earth 5 whenever we can find the absolute length of 
one degree on its surface. 

Co-Latitude. — Co-latitude is the distance, in degrees, of 
any place from the nearest pole. 

The latitude and co -latitude ( complement of the latitude ) 
must, of course, always make 90 degrees. 
Parallels Parallels of latitude are small circles on the surface of the 
of latitude, earth, parallel to the equator. 

Every point, in such a circle, has the same latitude. 
Longitude. — The longitude of a place, on the surface of 
the earth, is the inclination of its meridian to some other 
meridian which may be chosen to reckon from. English 
astronomers and geographers take the meridian which runs 
through Greenwich Observatory, as the zero meridian. 
The first Other nations generally take the meridian of their princi- 
meridian ar- p a } observatories, or that of the capital of their country, as 
the first meridian ; but this is national vanity, and creates 
only trouble and confusion ; it is important that the whole 
world should agree on some one meridian, from which to reckon 
longitude ; but as nature has designated no particular one, it 
is not wonderful that different nations have chosen different 
lines. 
We adopt j n this work, we shall adopt the meridian of Greenwich as 
of Green- ^ e zero ^ me °f longitude, because most of the globes and 
wich ; and maps, and all the important astronomical tables, are adapted 
to that meridian, and we see nothing to be gained by chang- 
ing them. 

Declination. — Declination refers only to the celestial equa- 
tor, and is a leaning or declining, north or south of that line, 
and is similar to latitude on the earth. 

Solstitial Points. — The points, in the heavens, north and 
south, where the sun has its greatest declination. 

The northern point we call the Summer Solstice, and the 
southern point the Winter Solstice; the first is in longitude 
90°, the other in longitude 270°. 

As latitude is reckoned north and south, so longitude is 



INTRODUCTION. 5 

reckoned east and west ; but it would add greatly to syste- introduc. 
matic regularity, and tend much to avoid confusion and am- improve- 
biguity in computations, were this mode of expression aban- ment sng- 
doned, and longitude invariably reckoned westward, from to s esied - 
360 degrees. 

Latitude and longitude, on the earth, does not corre- Latitude, 
spond to latitude and longitude in the heavens. Latitude, on OB § itn f e > 

*■ p and right as- 

the earth, corresponds with declination in the heavens ; and cension. 
longitude, on the earth, has a striking analogy to right ascen- 
sion in the heavens, though not an exact correspondence. 
We shall more particularly explain latitude, longitude, and 
right ascension in the heavens, as we advance in this work ; 
for it is only when we are forced to use these terms, that the 
nature and spirit of their import can be really understood. 

There are other technicalities, and terms of frequent use, other terms 
in astronomy, such as Conjunction, Opposition, Retrograde, not exp ain ' 
Direct, Apogee, Perigee, &c, &c, all of which, for the sake 
of simplicity, had better not be explained until they fall 
into use ; and, once for all, let us impress this fact on the 
minds of our readers, that we shall put far more stress on the 
substance and spirit of a thing, than on its name. 



6 ASTRONOMY. 



SECTION I. 



CHAPTER I. 

PRELIMINARY OBSERVATIONS. 

chap. i. To commence the study of astronomy, we must observe 
and call to mind the real appearances of the heavens. 

Take such a station, any clear night, as will command an 
extensive view of that apparent, concave hemisphere above 
us, which we call the sky, and fix well in the mind the direc- 
tions of north, south, east, and west. 
The appa- At first, let us suppose our observer to be somewhere in 
of the stars. * ne United States, or somewhere in the northern hemisphere, 
about 40 degrees from the equator. 

As yet, this imaginary person is not an astronomer, and 
neither has, nor knows how to use, any astronomical instru- 
ment ; but we would have him mark with attention the po- 
„ sitions of the heavenly bodies. 

( 1. ) Soon he will perceive a variation in the position of 
the stars ; those at the east of him will apparently rise ; those 
at the west will appear to sink lower, or fall below the hori- 
zon; those at the south, and near his zenith, will apparently 
move westward ; and those at the north of him, which he may 
see about half way between the horizon and zenith, w ill appear 
stationary. 
Apparent Let such observations be continued during all the hours 
revolution of Q ^ ^ Q night, and for several nights, and the observer cannot 

the heavenly ° ° 

bodies. fail to be convinced that not only all the stars, but the sun, 
moon, and planets, appear to perform revolutions, in about 
twenty-four hours, round a fixed point ; and that fixed point, 
as appears to us (in the middle and northern part of the 
United States ), is about midway between the northern hori- 
zon and the zenith. 
Large and ft g^o^d always be borne in mind, that the sun, moon, and 
s " stars, have an apparent diurnal motion round a fixed point. 



PRELIMINARY OBSERVATIONS. 7 

and all those stars which are 90 degrees from that point, Chap. i. 
apparently describe a great circle. Those stars that are 
nearer to the fixed point than 90 degrees, describe smaller 
circles ; and the circles are smaller and smaller as the objects 
are nearer and nearer the fixed points. 

( 2. ) There is one star so near this fixed point, that the 
small circle it describes, in about 24 hours, is not apparent 
from mere inspection. To detect the apparent motion of 
this star, we must resort to nice observations, aided by ma- 
thematical instruments. 

This fixed point, that we have several times mentioned, is The North 
the North Pole of the heavens, and this one star that we have just Star " 
mentioned, is commonly called the North Star, or the Pole Star. 

(3.) This star, on the 1st of January, 1820, was 1° 39' Position of 
6" from the pole, and on 1st of January, 1847, its distance * he North 
from the pole was 1° 30' 8"; and it will gradually and 
more slowly approach within about half a degree of the pole, 
and afterward it will as gradually recede from the pole, and 
finally cease to be the polar star. 

We here, and must generally, speak of the star, or the stars, The pole 
as in motion; but this is not so. The fixed stars are also- in motlon - 
lutely fixed ; it is the pole itself that has a slow motion among 
the stars, but the cause of this motion cannot now be ex- 
plained; it is one of the most abstruse points in astronomy, 
and we only mention it as a fact. 

As the North Star appears stationary, to the common ob- 
server, it has always been taken as the infallible guide to 
direction ; and every sailor of the ocean, and every wanderer 
of the African and Arabian deserts, has held familiar ac- 
quaintance with it. 

( 4. ) If our observer now goes more to the southward, and changes of 
makes the same observations on the apparent motions of the a PP earance 
stars, he will find the same general results ; each individual southward, 
star will describe the same circle ; but the pole, the fixed 
point, will be lower down, and nearer the northern horizon ; 
and it will be lower and lower in proportion to the distance 
the observer goes to the south. After the observer has gone 
sufficiently fax the fixed point, the pole, will no longer be up 
2 



8 ASTRONOMY. 

Chap. i. in the heavens, but down in the northern horizon ; and when 
Appear- the pole does appear in the horizon, the observer is at the 
ance from equator, and from that line all the stars at or near the equa- 
eqaa or. ^ a pp ear j. Q r j se U p (Ji rec tly from the east, and go down 
directly to the west; and all other stars, situated out of the 
equator, describe their small circles parallel to this perpendi- 
cular equatorial circle. 
south of j£ foe observer goes south of the equator, the apparent 
north pole of the heavens sinks below the northern horizon, 
and the south pole rises up into the heavens at the south. 
changes in /■ 5 \ jf t k e observer should go north, from the first 

appearance .., „ ,° , , , 

on going station, in place of going south, the north pole would rise 
north, nearer to the zenith; and, should he continue to go north, he 

would finally find the pole in his zenith, and all the stars 
would apparently make circles round the zenith, as a center, 
and parallel to the horizon ; and the horizon itself would be the 
celestial equator. 

( 6. ) When the north pole of the heavens appears at the 
zenith, the observer must then be at the north pole, on the 
earth, or at the latitude of 90 degrees. 
Appear- ( 7. ) Any celestial body, which is north of the equator, is 

ance from . . 

the north always visible from the north pole of the earth ; hence the 
pole. sun, which is north of the equator from the 20th of March to 

the 23d of September, must be constantly visible during that 
period, in a clear sky. 

Just as the sun comes north of the equator, its diurnal 
progress, or rather, the progress of 24 hours, is around the 
horizon. When the sun's declination is 10 degrees north of 
the equator, the progress of 24 hours is around the horizon 
at the altitude of 10 degrees ; and so for any other degree. 

From the north pole, all directions, on the surface of the 
earth, are south. North would be in a vertical direction 
toward the zenith. 
How to ^y e jj ave |3 Ser y e( i that the pole of the heavens rises as we 

rind the eir- L 

r.nmference go north, and sinks toward the horizon as we go south ; and 
andtMameter waen we observe that the pole has changed its position one 

of the earth. ox 

degree, in relation to the horizon, we know that we must have 
changed place one degree on the surface of the earth. 



PRELIMINARY OBSERVATIONS. 9 

( 8. ) Now we know by observation, that if we go north Chap, i 
about 69i English miles on the earth, the north pole will be 
one degree higher above the horizon. Therefore 69i miles 
corresponds to one degree, on the earth ; and hence the whole 
circumference of the earth must be 69^ multiplied by 360 : 
for there are 360 degrees to every circle. This gives 24,930 
miles for the circumference of the earth, and 7,930 miles for 
its diameter, which is not far from the truth. 

( 9. ) Here, in the United States, or anywhere either in Ch-cumpo- 
Europe, Asia, or America, north of the equator, say in lati- lar stars < 
tude 40°, the north pole of the heavens must appear at an 
altitude of 40° above the horizon ; and as all the stars and 
heavenly bodies apparently circulate round this point as a 
center, it follows that all those stars which are within 40° 
of the pole can never go below the horizon, but circulate 
round and round the pole. All those stars which never go 
below the horizon, are called circumpolar stars. 

At the north, and very near the north pole, the sun is a The sun a 
circumpolar body while it is north of the equator, and it is a ^ ir ^ um P° ar 

J- ^ u body, as seen 

circumpolar body as seen from the south pole, while it is south from the 
of the equator: this gives six months dav and six months nortn °f latl - 

..,,,; ° J tude CG de- 

night, at the poles. grees> 

( 10. ) North of latitude 66°, and when the sun's declina- 
nation is more than 23° north ( as it is on and about the 20th 
of June in each year ), then the sun comes at, or very near, the 
northern horizon, at midnight ; it is nearly east, at 6 o'clock 
in the morning ; it is south, at noon, and about 23° in alti- 
tude ; and is nearly west at 6 in the afternoon. 

( 11. ) In the southern hemisphere, there is no prominent 
star near the south pole ; that is, no southern polar star ; but, 
of course, there are circumpolar stars, and more and more as 
one goes south ; and if it were possible to go to the south 
pole, the whole southern hemisphere would consist of circum- 
polar stars, and the pole, or fixed point of the heavens, would 
be directly overhead ; and the sun himself, when south of the 
equator, would be a circumpolar body, going round and round 
every 24 hours ; nearly parallel with the horizon. 

( 12. ) In all latitudes, and from all places, the sun is 



10 ASTRONOMY. 

Chap. i. observed to circulate round the nearest pole, as a center ; and 

The near- when the sun is on the same side of the equator as the ob- 

est pole is gerverj more than half of the sun's diurnal circle is above the 

the center of. ... i i r» i 

the sun's di- horizon, and the observer -will have more than \A hours sun- 

nrnal mo- lio'ht. 

When the sun is on the equator, the horizon, of every lati- 
tude, cuts the sun's diurnal circle into two equal parts, and 
gives 12 hours day, and 12 hours night, the world over. 
When the sun is on the opposite side of the equator from the 
observer, the smaller segment of the sun's diurnal circle is 
above the horizon, and, of course, gives shorter days than 
nights. 

We have, thus far, made but rude and very imperfect ob- 
servations on the apparent motion of the heavenly bodies, and 
have satisfied ourselves only of two facts : 
Facts set- l. That all the stars, sun, moon, and planets included, 
apparently circulate round the pole, and round the earth, in 
a day, or in about 24 hours. 

2. That the sun comes to the meridian, at different alti- 
tudes above the horizon, at different seasons of the year, 
giving long days in June, and short days in December. 

(13.) Let us now pay attention to some other particulars. 

Let us look at the different groups of stars, and individual 

stars, so that we can recognize them night after night. 

Necessity We should now have some means of measuring time ; but, 

measure^of * n ear l y ^ a y s > when astronomy was no further advanced than 

time. it is supposed to be in this work, a clock could hardly have 

had existence; and the advancement of timepieces has been 

nearly as gradual as the advancement of astronomy itself. 

But we will not dwell on the history, and difficulties, of 
clockmaking; whatever these difficulties may have been, or 
whatever niceties modern science and art may have attained, 
there never was a period when people had not a good general 
idea of time, and some means to measure it. For instance, 
sunrise and sunset could be always noted as distinct points 
of time ; and the interval of a day and a night, or an astro- 
nomical day, which we now call 24 hours, was soon observed 
to be a constant quantity. 



PRELIM IN SHY OBSERVATIONS- H 

At first, only rude timepieces could be made, designed to Chap. i. 
mark off equal intervals of time; but we will suppose, at 
once, that the reader of this work, or our imaginary observer, 
can have the use of a common clock, which measures mean 
solar time of 24 hours in a natural day, which is marked by 
the sun. 

( 14.) Now, having power to recognize certain stars, or The parti- 
groups of stars, such as the Seven Stars, the Belt of Orion, cu ar _ pos 

o -t ' ' * ' tion of stars 

Aldebaran, Sirius, and the like, and having likewise the use in relation to 
of a clock, he can observe token any particular star comes to time * 
any definite position. 

Let a person place himself at any particular point, to the 
north of any perpendicular line, as the edge of a wall or 
building, and let him observe the stars as they pass behind 
the building, in their diurnal motions from the east to the 
west. For example, let us suppose that the observer is 
watching the star Aldebaran, and that, when the eye is placed 
in a particular definite position, the star passes behind the 
building at exactly 8 o'clock. 

The next evening, the same star will come to the same 
point about 4 minutes before 8 o'clock ; and it will not come 
to the same point again, at 8 o'clock in the evening, until 
after the expiration of one year. 

(15.) But in any year, on the same day of the month, and 
at the same hour of the day, the same star will be at, or very 
near, the same position, as seen from the same point. 

For instance, if certain stars come on the meridian at a on stars 
particular time in the evening, on the first day of December, comin s t0 

i -n -i .-,. . . the men- 

the same stars will not come on the meridian agam, at the d i an . 
same time of the night, until the first day of the next December. 

On the first of January, certain stars come to the meridian index to 
at midnight; and ( speaking loosely) every first of January thelen s thof 
the same stars come to the meridian at the same time ; and 
there will be no other day during the whole year, when the 
same stars will come to the meridian at midnight. 

Thus, the same day of every year is observed to have the 
same position of the stars at the same hour of the night ; and 
this is the most definite index for the expiration of a year. 



12 ASTRONOMY. 

Chap » r - ( 16.) The year is also indicated by the change of the sun's 

Another declination, which the most careless observer cannot fail to 

index of the notice< Qn the 21st of June, the sun declines about 23i de- 

jength of the 

year. grees from the equator toward the north ; and, of course, to 

us in the northern hemisphere, its meridian altitude is so 
much greater, and the horizontal shadows it casts from the 
same fixed objects will be shorter; and the same meridian 
altitude and short shadow will not occur again until the fol- 
lowing June, or after the expiration of one year. 

Thus, we see, that the time of the stars coming on to the 
meridian, and the declination of the sun, have a close corre- 
spondence, in relation to time. 
Fixed In all our observations on the stars, we notice that their 
thi^ternTis apparent relative situations are not changed by their diurnal 
applied, motions. In whatever parts of their circles they are observed, 
or at whatever hour of the night they are seen, the same con- 
figuration is recognized, although the same group, in the 
different parts of its course, will stand differently, in respect 
to the horizon. For instance, a configuration of stars resem- 
bling the letter A, when east of the meridian, will resemble 
the letter V, when west of the meridian. 
Wander- As the stars, in general, do not change their positions, in 
respect to each other, they are called fixed stars ; but there 
are a few important stars that do change, in respect to other 
stars ; and for that reason they become especial objects of 
attention, and form the most interesting portion of astro- 
nomy. 

In the earliest ages, those stars that changed their places, 
were called wandering stars ; and they were subsequently 
found to be the planetary bodies of the solar system, like the 
earth on which we live. 



Planets. 



APPEARANCES IN THE HEAVENS. 13 



CHAPTER II. 

APPEARANCES IN THE HEAVENS. 

In the preceding chapter we have only called to mind the chap, ii . 
most obvious and preliminary observations, which force them- 
selves on every one who pays the least attention to the 
subject. 

We shall now consider the observer at one place, making- 
more minute and scientific observations. 

( 17.) We have already remarked, that if the observer How to 
was on the equator, the poles, to him, would be in his horizon. ° ^ the 
If he were at one of the poles, for instance, the north pole, the place of ob- 
equator would then bound the horizon. If he were half way servatlon - 
between the equator and one of the poles, that pole would 
appear half way between the horizon and the zenith. 

Therefore, by observing the altitude of the pole above the hori- 
zon, we determine the number of degrees we are from the 
equator, which is called the latitude of the place. 

( 18.) To carry the mind of the reader progressively along, 
in astronomy, we must now suppose that he not only has the 
use of a good clock, but has also some instrument to measure 
angles. 

Clocks and astronomical instruments progressed toward 
perfection in about the same ratio as astronomy itself; but, 
as we are investigating or leading the young mind to the in- 
vestigation of astronomy, and not making clocks or mathe- 
matical instruments, we therefore suppose that the observer 
has all the necessary instruments at his command, and we 
may now require him to make a correct map of the visible 
heavens ; but to accomplish it, we must allow him at least 
one year's time, and even then he cannot arrive at anything 
like accuracy, as several incidental difficulties, instrumental 
errors, and practical inaccuracies, must be met and overcome. 

(19.) There are three principal sources of error, which Sources of 
must be taken into consideration, in making astronomical ^l* <&<&&*- 
observations. 1. Uncertainty as to the exact time. 2. Inex- tion- 



14 ASTRONOMY 

Chap. ii. pertness and want of tact in the observer ; and 3. Imperfec- 
tion in the instruments. Everything done by man is neces- 
m sarily imperfect. 
Practical " It may be thought an easy thing," says Sir John Her- 
and 011 ^^es sc ^ e ^ " ^°7 one unacquainted with the niceties required, to 
of error. turn a circle in metal, to divide its circumference into 360 
equal parts, and these again into smaller subdivisions, — to 
place it accurately on its center, and to adjust it in a given 
position ; but practically it is found to be one of the most 
difficult. Nor will this appear extraordinary, when it is con- 
sidered that, owing to the application of telescopes to the 
purposes of angular measurement, every imperfection of struc- 
ture or division becomes magnified by the whole optical power 
of that instrument ; and that thus, not only direct errors of 
workmanship, arising from unsteadiness of hand or imperfec- 
tion of tools, but those inaccuracies which originate in far 
more uncontrollable causes, such as the unequal expansion 
and contraction of metallic masses, by a change of tempera- 
ture, and their unavoidable flexure or bending by their own 
weight, become perceptible and measurable." 
Necessary ( 20.) The most important instruments, in an observatory, 
aside from the clock, are a circle, or sector, for altitudes; and 
a transit instrument. 

The former consists of a circle, or a portion of a circle, of 
firm and durable material, divided into degrees, at the rate 
of 360 to the whole circle. Each degree is divided into equal 
parts ; and, by a very ingenious mechanical adjustment of an 
index, called a Vernier scale, the division of the degree is 
practically (though not really) subdivided into seconds, or 
3600 equal parts. 

The whole instrument must now be firmly placed and ad- 
justed to the true horizontal ( which is exactly at right angles 
to a plumb line ), and so made as to turn in any direction. 
With this instrument we can measure angles of altitude. 

(21.) The transit instrument is but a telescope, firmlv fas- 
sit instru- ,-,,. ,. , i • i i 

ment# tened on a horizontal axis, east and west, so that the telescope 

itself moves up and down in the plane of the meridian, but can 
never be turned aside from the meridian to the east or west. 



The tran- 



APPEARANCES IN THE HEAVENS. 



15 




Transit Instrument. 




Meridian Wires. 



To place the instrument in this posi- 
tion, is a very difficult matter ; but it is 
a difficulty which, at present, should not 
come under consideration; we simply 
conceive it so placed, ready for observa- 
tions. 

" In the focus of the eyepiece, and at 
right angles to the length of the tele- 
scope, is placed a system of one horizontal and five equidis- 
tant vertical threads or wires, as represented in the annexed 
figure, which always appear in the field of view, when properly 
illuminated, by day by the light of the 
sky, by night by that of a lamp, intro- 
duced by a contrivance not necessary here 
to explain. The place of this system of 
wires may be altered by adjusting screws, 
giving it a lateral (horizontal) motion; 
and it is by this means brought to such a 
position, that the middle one of the vertical wires shall inter- 
sect the line of collimation of the telescope, where it is arrested 
and permanently fastened. In this situation it is evident 
that the middle thread will be a visible representation of that 
portion of the celestial meridian to which the telescope is 
pointed ; and when a star is seen to cross this wire in the 
telescope, it is in the act of culminating, or passing the celes- 
tial meridian. The instant of this event is noted by the 
clock vor chronometer, which forms an indispensable accom- 
paniment of the transit instrument. For greater precision, 
the moments of its crossing all the five vertical threads is 
noted, and a mean taken, which ( since the threads are equi- 
distant ) would give exactly the same result, were all the 
observations perfect, and will, of course, tend to subdivide and 
destroy their errors in an average of the whole." 

( 22. ) Thus, all prepared with a transit instrument and a 
clock, we fix on some bright star, and mark when it comes to 
the meridian, or appears to pass behind the central wire of the 
instrument. By noting the same event the next evening, the 
next, and the next, we find the interval to be very sensi- 



Chap, II. 



A line in 
the transit 
instrument a 
visible meri- 
dian. 



Practical 
artifices, to 
attain accu- 
racy. 



Intervals 
between the 
fixed stars 
passing the 
meridian al- 
ways con- 
stant. 



16 



ASTRONOMY. 



Chap. ii. bly less than 24 hours ; but the intervals are equal to each 
other : and all the fixed stars are unanimous in giving equal 
intervals of time between two successive transits of the same star, 
if measured by the same clock. 

The following observations were actually taken by M. 
Arago and Lacroix, in the small island of Formentera, in the 
Mediterranean, in December, 1807. 



Date of Observations. 


Time of transit of the 
Star a, Arietis. 


Intervals between 
successive Transits. 


1807, Dec. 24, 
" 25, 
" 26, 
" " 27 

" 28,' 


h. m. s. 
9 42 32.36 
9 41 29.70 
9 40 26.72 
9 39 23.90 
9 38 21.38 


h. m. s. 

23 58 57.34 
23 58 57.02 
23 58 57.18 
23 58 57.48 



of measure 
for time. 



These intervals between the transits agree so nearly, that 
it is very natural to suppose them exactly equal, and the 
small difference of the fraction of a second to arise from some 
slight irregularities of the clock, or imperfection in making 
the observations. 

The equality of these intervals is not only the same for all 
the fixed stars, in passing the meridian, but they are the 
same in passing all other planes. 
standard Now as this has been the universal experience of astrono- 
mers in all ages, it completely establishes the fact, that all 
the fixed stars come to the meridian in exactly equal inter- 
vals of time ; and this gives us a standard measure for time, 
and the only standard measure, for all other motions are 
variable and unequal. 
Time of Again, this interval must be the time that the earth 
the earth's em p] y S } n turning on its axis; for if the star is fixed, it is a 

revolution on x J . 

its axis. mark for the time that the meridian is in exactly the same 
position in relation to absolute space. 
M.Arago's ^ 23.) That the reader may not imbibe erroneous impres- 
sions, we remark, that the clock used for the preceding ob- 
servations, made by M. Arago and Lacroix, ran too fast, if it 
was a common clock, and too slow, if it was an astronomical 



clock. 



APPEARANCES IN THE HEAVENS. 17 

clock. It was not mentioned which clock was used, nor was Chap. n. 
it material simply 'to establish the fact of equal intervals ; nor 
was it essential that the clock should run 24 hours, in a mean 
solar day ; it was only essential that it ran uniformly, and 
marked off equal hours in equal times. 

If it had been a common clock, and ran at a 'perfect rate, 
the interval would have been 23 h. 56 m. 4.09 s. 

( 24.) In the preceding section we have spoken of an An astro- 
astronomical clock. Soon after the fact was established that B ^ ca 
the fixed stars came to the meridian in equal times, and that 
interval less than 24 hours, astronomers conceived the idea 
of graduating a clock to that interval, and dividing it into 24 
hours. Thus graduating a clock to the stars, and not to the 
sun, is called a sidereal, and not a solar, or common clock ; 
and as it was suggested by astronomers, and used only for 
the purposes of astronomy, it is also very appropriately called 
an astronomical clock; but save its graduation, and the 
nicety of its construction, it does not differ from a common 
clock. 

With a perfect astronomical clock, the same star will pass the To deter- 
meridian at exactly the same time, from one year's end to an- mmetherate 

. . . J . of an astro- 

other.* If the time is not the same, the clock does not run nomicai 

clock. 

* Sidereal time-has been slightly modified since the discovery of the 
precession of the equinoxes, though such modification has never been 
distinctly noticed in any astronomical work. 

At first, it was designed to graduate the interval between two suc- 
cessive transits of the same star over the meridian, to 24 hours, and to 
call this a sidereal day ; which, in fact, it is. 

But it was necessary, in some way, to connect sidereal with solar 
time ; and, to secure this end, it was determined to commence the side- 
real day (not from the passage of any particular star across the meri- 
dian, but from the passage of the imagi nary point in the heavens, where 
the sun's path crosses the vernal equinox, called the first point of 
Aries), thus making the sidereal day and the equinoctial year commence 
at the same moment of absolute time. 

For some time, it was supposed that the interval between two suc- 
cessive transits of the first point of Aries, over the m ridian, was the 
same as two successive transits of a star ; but the two intervals are not 
identical; the first point of Aries has a very slow motion westward 
among the stars, which is called the precession of the equinox, and 
2 B* 



18 ASTRONOMY. 

Chap. ii. to sidereal time ; and the variation of time, or the difference 
between the time when the star passes the meridian, and the 
time which ought to be shown by the clock, will determine 
the rate of the clock. And with the rate of the clock, and its 
error, we can readily deduce the true time from the time 
shown by the face of the clock. 

Solar days ( 25. ) When we examine the sun's passage across the 

equa ' meridian, and compare the elapsed intervals with the sidereal 

clock, we find regular and progressive variations, above and 

below a mean period, that cannot be accounted for by errors 

of observation. 

The mean interval, from one transit of the sun to another, 
or from noon to noon, when we take the average of the whole 
year, is 24 hours, of solar time, or 24 h. 3 m. 56.5554 s. of 
sidereal time ; but, as we have just observed, these intervals 
are not uniform; for instance, about the 20th of December, 
they are about half a minute longer, and about the 20th of 
September, they are as much shorter, than the mean period. 

The snn From this fact, we are compelled to regard the sun, not as 
must have a g xe( j point ; it must have motions, real or apparent, inde- 

real or appa- . . , 

rent motion, pendent of the rotation of the earth on its axis. 

( 26. ) When we compare the times of the moon passing 

the meridian, with the astronomical clock, we are very forcibly 

struck with the irregularity of the interval. 

General The least interval between two successive transits of the 

motion of moon / W nich may be called a lunar day ), is observed to be 

the moon. v J _ 

about 24 h. 42 m. ; the greatest, 25 h. 2 m.; and the mean, or 
average, 24 h. 54 m., of mean solar time. 

These facts show, conclusively, that the moon is not a 

which makes its transits across the meridian a fraction of a second 
shorter than the transits of a star. 

The time required for 366 transits of a star across the meridian, is 
( 3". 34), three seconds and thirty-four hundredths of a second of sidereal 
time, greater than for 366 transits of the equinox. 

This difference would make a day in about 25000 years. The time 
elapsed between two successive transits of the equinox being now 
called a sidereal day of ----- 24h. m. s., the 
time between the transits of the same star, is - 24 h. m. 0.00916 s 

Every astronomer understands Art. ( 24 ) with this modification. 



APPEARANCES IN THE HEAVENS. 19 

fixed body, like a fixed star, for then the interval would be chap. ii. 
24 hours of sidereal time. 

But as the interval is always more than 24 hours, it shows 
that the general motion of the moon is eastward among the 
stars, with a daily motion varying from 10^- to 16 degrees* 
traveling, or appearing to travel, through the whole circle 
of the heavens ( 360° ) in a little more than 27 days. 

Thus, these observations, however imperfectly and rudely chief ob- 
taken, at once disclose the important fact, that the sun and jec 
moon are in constant change of position, in relation to the 
stars, and to each other ; and, we may add, that the chief 
object and study of astronomy, is, to discover the reality, the 
causes, the nature, and extent of such motions. 

(27.) Besides the sun and moon, several other bodies 0ther 

.'."-, . ,-. . -,. ■■ . movable and 

were noticed as coming to the meridian at very unequal m- wandering 
tervals of time — intervals not differing so much from 24 bodies, 
sidereal hours as the moon, but, unlike the sun and moon, 
the intervals were sometimes more, sometimes less, and some- 
times equal to 24 sidereal hours. 

These facts show that these bodies have a real, or appa- 
rent motion, among the stars, which is sometimes westward, 
sometimes eastward, and sometimes stationary; but, on the 
whole, the eastward motion preponderates ; and, like the sun 
and moon, they finally perform revolutions through the hea- 
vens from west to east. 

Only four such bodies ( stars ) were known to the ancients, Wandering 
namely, Venus, Mars, Jupiter, and Saturn. stars known 

These stars are a portion of the planets belonging to our cients. 
solar system, and, by subsequent research, it was found that Modem 
the Earth was also one of the number. As we come down 
to more modern times, several other planets have been disco- 
vered, namely, Mercury, Uranus, Vesta, Juno, Ceres, Pallas, 
and, very recently ( 1846), the planet Neptune. \ 

* Four minutes above 24 hours corresponds to one degree of arc. 

t We have not mentioned the names of these planets in the order in 
which they stand in the system, but rather in the order of their dis- 
covery. As yet, we have really no idea of a planet, or a planetary 
system. 



de nee 
Uveen 



20 ASTRONOMY. 

Chap. ii. We shall again examine the meridian passages of the sun, 
moon, and planets, and deduce other important facts con- 
cerning them, besides that of their apparent, or real motions 
among the fixed stars. 
Observa- (28.) But let us return to the fixed stars. We have 

. f ~ several times mentioned the fact, that the same star returns 

determine » 

the meridian to the same meridian again and again, after every interval of 
distances of 24 s j(j erea i hours. So two different stars come to the meri- 

tne stars. 

dian at constant and invariable intervals of time from each 
other ; and by such intervals we decide how far, or how many 
degrees, one star is east or west of another. For instance, 
if a certain fixed star was observed to pass the meridian when 
the sidereal clock marked 8 hours, and another star was ob- 
served to pass at 9, just one sidereal hour after, then we 
know that the latter star is on a celestial meridian, just 15 
degrees eastward of the meridian of the first mentioned star. 
Correspou- As 24 hours corresponds to the whole circle, 360 degrees, 
be ' therefore one hour corresponds to 15 degrees ; and 4 minutes, 
and decrees. i* 1 time, to one degree of arc. Hence, whatever be the ob- 
served interval of time between the passing of two stars over 
the meridian, that interval will determine the actual difference 
of the meridians running through the stars ; and when we 
know the position of any one, in relation to any celestial meri- 
dian, we know the positions of all whose meridian observations 
have been thus compared. 
Right as- The position of a star, in relation to a particular celestial 
meridian, is called Right Ascension, and may be expressed 
either in time or degrees. Astronomers have chosen that 

It is true, we might mention every fact, and every particular re- 
specting each planet ; such as its period of revolution, size, distance 
from the sun, &c. ; but such facts, arbitrarily stated, would not convey 
the science of astronomy to the reader, for they can be told alike to the 
man and to the child — to the intellectual and to the dull — to the learned 
and to the unlearned. 

To constitute true knowledge — to acquire true science — the pupil 
must not only know the fact, but how that fact was discovered, or de- 
duced from other facts. Hence we shall mainly construct our theories 
from observations, as we pass along, and teach the pupil to decide the 
case from the facts, evidences, and circumstances presented. 



cension. 



REFRACTION. 21 

meridian, for the first meridian, which passes through the Chap. n. 
sun's center at the instant the sun crosses the celestial equa- First men- 
tor in the spring, on the 20th of March. dian - 

Right ascension is measured from the first meridian, east- 
ward, on the equator, all the way round the circle, from to 
360 degrees, or from Oh. to 24 h. 

The reason why right ascension is not called longitude will 
be explained hereafter. 

(29.) If we observe and note the difference of sidereal To find the 
time between the coming of a star to the meridian, and the " g ^ C T' 

© . ' sions of the 

coming of any other celestial body, as the sun, moon, planet, sun, moon, 
or comet, such difference, applied to the right ascension of the and planets - 
star, will give the right ascension of the body. 

But every astronomer regulates, or aims to regulate, his 
sidereal clock, so that it shall show Oh. m. s., when the 
equinox is on the meridian ; and, if it does so, and runs regu- 
larly, then the time that anybody passes the meridian by the 
clock, will give the right ascension of the body in time, with- 
out any correction or calculation; but, practically, this is 
never the case ; a clock is never exact, nor can it ever run 
exactly to any given rate or graduation. 

We have thus shown how to determine the right ascensions 
of the heavenly bodies. We shall explain how to find their 
positions in declination, in the next chapter. 



CHAPTER III. 



REFRACTION. POSITION OF THE EQUINOX, AND OBLIQUITY OF 

THE ECLIPTIC HOW FOUND BY OBSERVATION. 

( 30. ) To determine the angular distance of the stars from Chap. hi. 
the pole, the observer must first know the distance of his 
zenith from the same point. 

As any zenith is 90 degrees from the true horizon, if the 
observer can find the altitude of the pole above the horizon 



22 



ASTRONOMY. 



by original 

observa 

tions, 



The mural 
circle. 



chap. hi. ( w hich is the latitude of the place of observation ), he, of 

course, knows the distance between the zenith and the pole. 

Prepara- ^ s fas north pole is but an imaginary point, no star being 

t^mming 6 " there, we cannot directly observe its altitude. But there is a 

the latitude ver y bright star near the pole, called the Polar Star, which, 

as all other stars in the same region, apparently revolves 

round the pole, and comes to the meridian twice in 24 sidereal 

hours; once above the pole, and once below it; and it is 

evident that the altitude of the pole itself must be midway 

between the greatest and least altitudes of the same star, 

provided the apparent motion of the star round the pole is really 

in a circle ; but before we examine this fact, we will show how 

altitudes can be taken by the mural circle. 

(31.) The mural, or 
wall circle, is a large me- 
tallic circle, firmly fas- 
tened to a wall, so that 
its plane shall coincide 
with the plane of the me- 
ridian. 
Ji A perpendicular line 
through the center, ZJV, 
(Fig. 2), represents the 
zenith and nadir points ; 
and at right angles to 
this, through the center, 
is the horizontal line, Hh. 
A telescope, Tt, and an index bar, Ii, at right angles to 
the telescope, are firmly fixed together, and made to revolve 
on the center of the mural circle. 

The circle is graduated from the zenith and nadir points, 
each way, to the horizon, from to 90 degrees. 

When the telescope is directed to the horizon, the index 
points, I and i, will be at Z and N", and, of course, show 0° 
of altitude. When the telescope is turned perpendicular, to 
Z, the index bar will be horizontal, and indicate 90 degrees 
of altitude. 

When the telescope is pointed toward any star, as in the 




How to ob- 
serve meri- 
dian alti- 
tudes. 



REFRACTION. 23 

figure, the index points, / and i, will show the position of the chap. hi. 
telescope, or its angle from the horizon, which is the altitude 
of the star. 

As the telescope, and index of this instrument, can revolve Mural ek. 
freelv round the whole circle, we can measure altitudes with cle a!s0 . a 

J transit m- 

it equally well from the north or the south; but as it turns strument. 
only in the plane of the meridian, we can observe only meri- 
dian altitudes with it. 

This instrument has been called a transit circle, and, says 
Sir John Herschel, " The mural circle is, in fact, at the same 
time, a transit instrument; and, if furnished with a proper 
system of vertical wires in the focus of its telescope, may be 
used as such. As the axis, however, is only supported at one 
end, it has not the strength and permanence necessary for 
the more delicate purposes of a transit ; nor can it be veri- 
fied, as a transit may, by the reversal of the two ends of its 
axis, east for west. Nothing, however, prevents a divided 
circle being permanently fastened on the axis of a transit 
instrument, near to one of its extremities, so as to revolve 
with it, the reading off being performed by a microscope 
fixed on one of its piers. Such an instrument is called a 
transit circle, or a meridian circle, and serves for the simulta- 
neous determination of the right ascensions and polar dis- 
tances of objects observed with it ; the time of transit being 
noted by the clock, and the circle being read off by the late- 
ral microscope." 

( 32.) To measure altitudes in all directions, we must have Altitude 

,i • , yn ,• r- .i ' and azimuth 

another instrument, or a modification of this. instrnm 

Conceive this instrument to turn on a perpendicular axis, 
parallel to Z N, in place of being fixed against a wall ; and 
conceive, also, that the perpendicular axis rests on the center 
of a horizontal circle, and on that circle carries a horizontal 
index, to measure azimuth angles. 

This instrument, so modified, is called an altitude and azi- 
muth instrument, because it can measure altitudes and azi- 
muths at the same time. 

( 83.) After astronomy is a little advanced, and the angu- 
lar distance of each particular star, sun, moon, and planet, 
3 



24 ASTRONOMY. 

Chap. hi. from the pole is known, then we can determine the latitude by 
The lati- observing the meridian altitude of any known celestial body ; 
tnde taken ^^ ^f^g ^heir positions are established ( as is now supposed 
tude of the to be the case with the reader of this work ), the only way to 
pole " observe the latitude is by the altitudes of some circumpolar 

star, as mentioned in Art. 30. 

To settle this very important element, the observer turns 
the telescope of his mural circle to the pole star, and ob- 
serves its greatest and least altitudes, and takes the half sum 
for his latitude. But is this really his latitude ? Does it 
require any correction, and if so, what, and for what reason V 
a difficulty. At first, it was very natural to suppose that this gave the 
exact latitude; but astronomers, ever suspicious, chose to 
verify it, by taking the same observations on other circum- 
polar stars ; and if the theory was correct, and the observa- 
tions correctly taken, all circumpolar stars would give the 
same, or very nearly the same, result. Such observations 
were made, and stars at the same distance from the pole, 
gave the same latitude, and stars at different distances from 
the pole, gave different latitudes ; and the greater the dis- 
tance of any star from the pole, the greater the latitude de- 
duced from it. A star 30 or 35 degrees from the pole, ob- 
served from about the latitude of 40 degrees, will give the 
latitude 12 or 15 minutes of a degree greater than the pole 
star. 
New and Astronomers were now troubled and perplexed. These 
truths ant g rea t an( ^ manifest discrepancies could not be accounted for 
by imperfection of instruments, or errors of observations, and 
some unconsidered natural cause was sought for as a solution. 
Curves de- T/o bring more evidence to bear on the case, astronomers 
circumpoia/ examined the apparent paths of the stars round the pole, by 
stars. means of the altitude and azimuth instrument, and they were 

found to be not exact circles ; but departed more and more 
from a circle, as the star was a greater and greater distance 
from the pole. 

These curves were found to be somewhat like ovals — the 
longer diameter passing horizontally through the pole — ■ the 



ASTRONOMICAL REFRACTION. 25 

upper segments very nearly semicircles, and the lower segments Chaf. hi. 
flattened on their under sides. 

With such evidences before the mind, men were not long 
in deciding that these discrepancies were owing to 



fraction. 



ASTRONOMICAL REFRACTION. 

( 34. ) It is shown, in every treatise on natural philosophy, General 
that light, passing obliquely from a rarer medium into a el 
denser, is bent toward a perpendicular to the new medium. 

Now, when rays of light pass, or are conceived to pass, 
from any celestial object, through the earth's atmosphere to 
an observer, the rays must be bent downward, unless they pass 
perpendicularly through the atmosphere ; that is, come from 
the zenith. 

EF,kz (Fig.' 
3 ), represent 
different strata 
of the earth's at- 
mosphere. Let 
s be a star, and 
conceive a line 
of light to pass 
from the star 
through the va- 
rious strata of 
air, to the ob- 
server, at 0. 

When it meets the first strata, as E F, it is slightly bent Refraction 
downward, and as the air becomes more and more dense, its increases al - 

/> ,. , titudes. 

retracting power becomes greater and greater, which more 
and more bends the ray. But the direction of the ray, at 
the point where it meets the eye of the observer, will deter- 
mine the position of the star as seen by him. Hence the 
observer at 0, will see the star at s', when its real position is 
at .5. 

As a ray of light, from any celestial object, is bent down- 




26 ASTRONOMY. 

Chap. hi. ward, therefore, as we may see by inspecting the figure, the 
altitude of all the heavenly bodies is increased by refraction. 

This shows that all the altitudes, taken as described in 
Art. 33, must be apparent altitudes — greater than true alti- 
tudes — and the resulting latitudes, deduced from them, all 
too great. 

The object is now to obtain the amount of the refraction 
corresponding to the different altitudes, in order to correct or 
attoio for it. 

To determine the amount of refraction, we must resort 
to observations of some kind. But what sort of observations 
will meet the case ? 
How to Conceive an observer at the equator, and when the sun or 

find the a- .. ,..,., 

monnt of re- a s * ar passes through, or very near his zenith, it has no re- 
fraction cor- fraction. But, at the equator, the diurnal circles are per- 
t^eve/'lie P en dicular to the horizon; and those stars which are very 
gree of aiti- near the equator, really change their altitudes in proportion to 
tude - the time. 

Now a star may be observed to pass the zenith, at the 
equator, at a particular moment : four hours afterward ( side- 
real time ), the zenith distance of this star must be 4 times 15, 
or 60 degrees, and its altitude just 30 degrees. But, by ob- 
servation, the altitude will be found to be 30° 1' 38". From 
this, we perceive, that V 38" is the amount of refraction 
corresponding to 30 degrees of altitude. 

In six sidereal hours from the time the star passed the 

zenith, the true position of the star would be in the horizon ; 

but, by observation, the altitude would be 33' 0", or a little 

more than the angular diameter of the sun. 

Amount From this, we perceive, that 33' 0" is the amount of re- 

of horizontal n . . . . 1 ■, 

refraction, faction at the horizon. 

Thus, by talcing observations at all intervals of time, between 
the zenith and the horizon, we can determine the refraction corre- 
sponding to every degree of altitude. 

( 35. ) In the last article, we carried the observer to the 
equator, to make the case clear ; but the mathematician need 
not go to the equator, for he can manage the case wherever 



ASTRONOMICAL REFRACTION. 27 

he may be — lie takes into consideration the curves, as men- chap. hi. 
tioned in Art. 33. 

If it were not for refraction, the curves round the pole Themathe- 
would be perfect circles, and the mathematician, by means of matlcian ' s 

1 method of 

the altitude and azimuth, which can be taken at any and nn ai ng t he 
every point of a curve, can determine how much it deviates amount of re- 
from a circle, and from thence the amount of refraction, or 
nearly the amount of refraction, at the several points. 

By using the refraction thus imperfectly obtained, he can 
correct his altitudes, and obtain his latitude, to considerable 
accuracy. Then, by repeating his observations, he can fur- 
ther approximate to the refraction. 

In this way, by a multitude of observations and computa- 
tions, the table of refraction ( which appears among the tables 
of every astronomical work ) was established and drawn out. 

( 36. ) The effect of refraction, as we have already seen, is Refraction 
to increase the altitude of all the heavenly bodies. There- ' ncreas ^ s 

J time ot sun- 

fore, by the aid of refraction, the sun rises before it otherwise light. 

would, and does not set as soon as it would if it were not 
for refraction ; and thus the apparent length of every day is 
increased by refraction, and more than half of the earth'' 8 sur- 
face is constantly illuminated. The extra illumination is equal 
to a zone, entirely round the earth, of about 40 miles in 
breadth. 

As the refraction in the horizon is about 33' of a degree, 
the length of a day, at the equator, is more than four minutes 
longer than it otherwise would be, and the nights four minutes 
less. 

At all other places, where the diurnal circles are oblique 
to the horizon, the difference is still greater, especially if we 
take the average of the whole year. 

In high northern latitudes, the long days of summer are Effects in 
very materially increased, in length, by the effects of refrac- hi s h lati - 
tion ; and near the pole, the sun rises, and is kept above the 
horizon, even for days, longer than it otherwise would be, 
owing to the same cause. 

Refraction varies very rapidly, in its amount, near the hori- 



28 ASTRONOMY. 

Chap. iii. zon ; and this causes a visible distortion of "both sun and 

moon, just as they rise or set. 
Distortion ;p or i ns t a nce, when the lower limb of the sun is just in the 
and moon in horizon, it is elevated, by refraction, 33'. 
the horizon. But the altitude of the upper limb is then 32', and the 
refraction, at this altitude, is 27' 50", elevating the upper 
limb by this quantity. Hence, we perceive, that the lower 
limb is elevated more than the upper ; and the perpendicular 
diameter of the sun is apparently shortened by 5' 10", and 
the sun is distinctly seen of an oval form; which deviates 
more from a circle below than above. 
An optical j/j^ apparently dilated size of the sun and moon, when 
near the horizon, has nothing to do with refraction : it is a 
mere illusion, and has no reality, as may be known by apply- 
ing the following means of measurement. 

Roll up a tube of paper, of such a size and dimensions as 
just to take in the rising moon, at one end of the tube, when 
the eye is at the other. After the moon rises some distance 
in the sky, observe again with this tube, and it will be found 
that the apparent size of the moon will even more than fill it. 

The reason of this illusion is well understood by the stu- 
dent of philosophy; but we are now too much engaged with 
realities to be drawn aside to explain illusions, phantoms, or 
any Will-o'-the-wisp. 

When small stars are near the horizon, they become invi- 
sible ; either the refraction enfeebles and dissipates their light, 
or the vapors, which are always floating in the atmosphere, 
serve as a cloud to obscure them. 
Application ( 37.) Having shown the possibility of making a table of 
' refraction corresponding to all apparent altitudes, we can now, 
by applying its effects to the observed altitudes of the cir- 
cumpolar stars, obtain the true latitude of the place of obser- 
vation. 

Let it be borne in mind, that the latitude of any place on 
the earth, is the inclination of its zenith to the plane of the 
equator ; which inclination is equal to the altitude of the pole 
above the horizon. 

We demonstrate this as follows. Let E ( Fig. 4 ) repre- 




ASTRONOMICAL REFRACTION. 29 

sent the earth. Fig. 4. Chap# iil 

Now, as an oh- ^> — i — ^^ T^n- 

server always con- y^ ^\ stration. 

ceives himself to 

be on the topmost 

part of the earth, 

the vertical point, 

Z, truly and natu- h 

rally represents his 

zenith. Through E, draw HE 0, at right angles to E Z ; 

then HE will represent the horizon ( for the horizon is 

always at right angles to the zenith). 

Let E Q represent the plane of the equator, and at right 
angles to it, from the center of the earth, must be the earth's 
axis ; therefore, E P, at right angles to E Q, is the direction 
of the pole. 

Now the arcs, - - ZP+P 0=90°, 

Also, - - - ZP+ZQ=90°, 



By subtraction, - P O—ZQ=0 ; 

Or, by transposition, the arc PO = ZQ; that is, the 
altitude of the pole is equal to the latitude of the place ; 
which was to be demonstrated. 

In the same manner, we may demonstrate that the arc, 
H Q, is equal to the arc Z P ; that is, the polar distance of 
the zenith is equal to the meridian altitude of the celestial equa- 
tor. Now, we perceive, that by knowing the latitude, we 
know the several divisions of the celestial meridian, from the 
northern to the southern horizon, namely, OP, P Z, Z Q, 
and QH 

( 38.) We are now prepared to observe and determine the 
declinations of the stars. 

The declination of a star, or any celestial object, is its meri- Deciina 
dian distance from the celestial equator. tl0n defined - 

This corresponds with latitude on the earth, and declination 
might have been called latitude. 

The term latitude, as applied in astronomy, is to be de- 
fined hereafter. 



30 ASTRONOMY. 

Chap. in. To determine the declination of a star, we must observe 

How to its meridian altitude ( "by some instrument, say the mural 

find the de- q^qU Fig. 2 ), and correct the altitude for refraction ( see 

clmationofa >-©••> ^ # \ 

star. table of refraction ) ; the difference will be the star's true 

altitude. 

If the true meridian altitude of the star is less than the meri- 
dian altitude of the celestial equator, then the declination of the 
star is south. If the meridian altitude of the star is greater 
than the meridian altitude of the equator, then the declination of 
the star is north. 

These truths will be apparent by merely inspecting Fig. 4. 

EXAMPLES. 

Examples 1. Suppose an observer in the latitude of 40° 12' 18" 
thod pursued nor ^ n ' observes the meridian altitude of a star, from the 
to find any southern horizon, to be 31° 36' 37" ; what is the declination 

star's <iecli- of thatstar? ^ 



nation. 



From • 90° 0' 00" 

Take the latitude, - 40 12 18_ 

Biff, is the meridian alt. of the equator, 49° 47' 42" 

Alt. of star, 31° 36' 37" 
Refraction, 1 32 

True altitude, 31° 35^ 5" - - 31° 35' 5" 

Declination of the star, south, - - 18° 12' 37" 

2. The same observer finds the meridian altitude of an- 
other star, from the southern horizon, to be 79° 31' 42"; 
what is the declination of that star ? 

Observed altitude, 
Refraction, - 

True altitude, - 

Altitude of equator, - 

Star's declination, north, - - - 29° 43' 49" 

3. The same observer, and from the same place, finds the 
meridian altitude of a star, from the northern horizon, to be 
51 c 29' 53"; what is the declination of that star? 



79° 


31' 


42" 
11 


79 


31 


31 


49 


47 


42 



51° 


29' 


53" 
46 


Chap, in, 






51 

40 


29 
12 


IT 
i 

18 




11 

78° 


16 

43' 


49 
11" 





ASTRONOMICAL REFRACTION. 31 

Observed altitude, 
Refraction, - 

True altitude of star, 
Altitude of pole ( or latitude ), 

Star from the pole ( or polar dist. ), 
Polar dist., from 90°, gives decl., north, 

In this way the declination of every star in the visible 
heavens can be determined. 

(39.) In Art. 28 we have explained how to obtain the Elements 

2; . , "; for a chart 

difference of the right ascensions of the stars ; and m the last of the stars# 
article we have shown how to obtain their declinations. 

With the declinations and differences of right ascensions, we may 
mark down the positions of all the stars on a globe or sphere — 
tlie true representation of the appearance of the heavens. 

Quite a region of stars exists around the south pole, which 
are never seen from these northern latitudes ; and to observe 
them, and define their positions, Dr. Halley, Sir John Her- 
schel, and several other English and French astronomers, 
have, at different periods, visited the southern hemisphere. 
Thus, by the accumulated labors of the many astronomers, 
we at length have correct catalogues of all the stars in both 
hemispheres, even down to many that are never seen by the 
naked eye. 

(40.) In Art. 28, we have explained how to find the dif- The zero 

. . meridian of 

ferences of the right ascensions of the stars ; but we have not right ascen- 
yet found the absolute right ascension of any star, for the want sioa - 
of the first meridian, or zero line, from which to reckon. But 
astronomers have agreed to take that meridian for the zero 
meridian, which passes through the sun's center the instant 
the sun comes to the celestial equator, in the spring ( which 
point on the equator is called the equinoctial point ) ; hut the 
difficulty is to find exactly where ( near what stars ) this mendian 
line is. Before we can define this line, we must take obser- 
vations on the sun, and determine where it crosses the equa- 
tor, and from the time we can determine the place. But be- 
fore we can place much reliance on solar observations, we 
must ask ourselves this question. Has the sun any parallax ? 



32 A S T R N M Y. 

Chap. in. that is, is the position of the sun jztst where it appears to be? 
Is it really in the plane of the equator, when it appears to be 
there ? 
Parallax. rp a ]} northern observers, is not the sun thrown back on the 
face of the sky, to a more southern position than the one it 
really occupies? Undoubtedly it is; and this change of 
position, caused by the locality of the observer, is called paral- 
lax ; but, in respect to the sun, it is too small to be considered 
in these primary observations. 

The early astronomers asked themselves these questions, 
and based their conclusions on the following consideration : 
Sun's pa- If the sun is materially projected out of its true place ; if it is 
raiiax msen- thrown to the southward, as seen by a northern observer, it 

sible, in com- # " 

monobserva- will cross the equator in the spring sooner than it appears 

tions. jj cr0SS- 

But let an observer be in the southern hemisphere, and, to 

him, the sun would be apparently thrown over to the north, 

and it would appear to cross the equator before it really did 

cross. Hence, if the sun is thrown out of place by parallax, 

an observer in the southern hemisphere would decide that the 

sun crossed the equator quicker, in absolute time, than that 

which would correspond to northern observations. 

Northern ~But, j n bringing observations to the test, it was found that 

observations both northern and southern observers fixed on the same, or 

compared, very nearly the same, absolute time for the sun crossing the 

equator. This proves that the position of the sun was not 

sensibly affected by parallax. 

We will now suppose (for the sake of simplicity) that a 
sidereal clock has been so regulated as to run to the rate of 
sidereal time ; that is, measure 24 hours between any two 
successive transits of the same star, over the same meridian, 
but the sidereal time not known. 

Also, suppose that, at the Observatory of Greenwich, in 
the year 1846, the following observations were made:* 

* In early times, such observations were often made. We took these 
results from the Nautical Almanac, and called them observations ; but, 
for the purpose of showing principles, it is immaterial whether obser- 
vations are real or imaginary. 



EQUINOCTIAL POINT 



33 



Date. 


Face of the Side- 


Declination by Observa. 


real Clock. 




( Art. 38. ) 




h. m. s. 


o 


/ // 


March 18, { 


1 3 20.00 





58 53.4 south, 


" 19, 


1 6 58.62 





35 11.3 " 


" 20, 


1 10 37.10 





11 29.4 " 


" 21, 


1 14 15.47 





12 12.0 north, 


" 22, 


1 17 54.07 





35 52:0 " 



Chap. III. 

Observa- 
tions to find 
the equinox, 
and the side- 
real time. 



From these observations, it is required to determine the sidereal 
time, or the error of the clock ; the time that the sun crossed the 
equator ; the sun's right ascension ; its longitude, and the obli- 
quity of the ecliptic. 

It is understood that the observations for declinations must 
have been meridian observations, and, of course, must have 
been made at the instant of apparent noon, local solar time. 

By merely inspecting these observations, it will be perceived 
that the sun must have crossed the equator between the 20th 
and 21st ; for at the apparent noon of the 20th, the declina- 
tion was 11' 29". 4 south ; and on the 21st, at apparent noon, 
it was 12' 12" north. Between these two observations, the 
clock measured out 24 h. 3 m. 38.37 s., of sidereal time. 

If the sun had not changed its meridian among the stars, 
the time would have been just 24 hours. The excess 
(3 m. 38.37 s.) must be changed into arc, at the rate of four 
minutes to one degree. Hence, to find the arc, we have this 
proportion : 

As 4 m "-. 3 m - 38.37 s - : : 1° : to the required result. 

The result is 54' 35". 4; the extent of arc which the sun 
changed right ascension during the interval between noon and 
noon of the 20th and 21st of March. 

To examine this matter understandingly, draw a line, E Q, 
( Fig. 5 ), and make it equal to 54' 35".4. 

From E, draw E S, at right angles to E Q, and make it computa- 
equal to 11' 29".4. From Q, draw QJV, at right angles to tions t0 find 
E Q, and make it equal to 12' 12". Then S will represent 
the sun at apparent noon, March 20th, and JV the position of 
the sun at apparent noon, on the 21st, and /S'iVis the line of 
3 



34 



ASTRONOMY. 



Ckaf. IIL 



Fig. 5. 




Error of 
the clock. 



Sun's right 
ascension. 



the sun among 
the stars, and 
the point op 
( called the 
first point of 

E Aries ), and it 
is where the 
sun crosses 

g the equator. 

Now we 
wish to find 
P where the line 
E Q is crossed by the line SIT; or, the object is, to find 
E op, expressed in time. 

To facilitate the computation, continue E S to P, making 
SP=zQJV, and draw the dotted line P Q. Then S P Q XT 
is a parallelogram. EP=1V 29".4+12' 12"==23' 41" A; 
and the two triangles, P E Q, and SE °p , are similar; there- 
fore we have 

PE : EQ : : SE : E<y. 

To have the value of E t, in time, E Q must be taken in 
time ; which is 3 m. 38.37 s. 

Hence, (23'41".4) : (3™- 38.37 s -) : IV 29" A : Ecp ; 

The result gives, Eop=l™- 45.91 s - 

But the clock time that the point E passed the meridian, 

was lh. 10 m. 37.10 s. 

Add, ----- 1 45.91 

The equi. passed merid. (by clock) at 1 h. 12 m. 23.01 
But, at the instant that the equinox is on the meridian, 

the sidereal clock ought to show Oh. m. s. 

The error of the clock was, therefore, 1 h. 12 m. 23.01 s. 

( subtractive ). 

As the whole line, EQ (in time ), is - 3 m. 38.37 s. 
And the part E op is - - - 1 45.91 

Therefore, qp Q, is - - - - 1 m. 52.46 
But op Q i s the right ascension of the sun at apparent noon, 



EQUINOCTIAL POINT. 35 

at Greenwich, on the 21st of March, 1846; a very important chap. in. 
element. , 

The right ascension of any heavenly body, whether it be How to 
sun, moon, star, or planet, is the true sidereal time that it find the ab " 

•t n l l r» solute right 

passes the meridian ; and now, as we have the error of the ascension or 
clock, we can determine the true sidereal time that any star the stars . 
passes the meridian, and, of course, its right ascension ; thus, an ^' lane ^ 
for example, 

If a star passed the meridian at - 10 h. 15 m. 47 s. 

Error of the clock is (subtractive) 1 12 23 

Eight ascension of the star is - 9 h. 3 m. 24 s. 

( 42.) To find the Greenwich apparent time, when the sun 
crossed the equinox, we refer to Fig. 5 ; and as the point E 
corresponds to apparent noon, of March 20th, and the Q to 
apparent noon of March 21st, and supposing the motion of 
the sun uniform (as it is nearly ) far that short interval, we 
have the following proportion : 

EQ : Hop : : 24h. : x. 

Giving to EQ and E<v their numeral values in seconds of 
sidereal time, the proportion becomes: 

218".37 : 105".91 : : 24h. : x. 

The result of this proportion gives 11 h. 38 m. 24 s., for the Time of 
interval, after the noon of the 20th of March, when the sun the e< i llmox - 
crossed the equator. 

This result is in apparent time. The difference between 
apparent time, and mean clock time, will be explained here- 
after. At this period, the difference between the sun and the 
common clock was 7 m. 36 s., to be added to apparent time. 

Equinox of 1846, March - - 20 d. 11 h. 38m. 24s. 
Equation of time (add), - 7 36 

Equinox, clock time (Greenwich), 20 d. 11 h. 46 m. 
(43.) The two triangles, E S°p and ^pQJV, are really obliquity 
spherical triangles ; but triangles on a sphere whose sides are of the eciip- 
less than a degree may be regarded as plane triangles, with- fou ' nd 
out any appreciable error. In the triangle E S^p, 
^V=1588".65, ES=6S9"A; 



36 ASTRONOMY 

Chap, iii . and, if we regard these seconds of arc as mere numerals, and 
calculate the angle E T S, we find it 23° 27' 43" ; which is 
the obliquity of the ecliptic. 
Sun's ion- By computing the length of the line S N, we find it 59' 30" ; 
which was the variation in the sun's longitude, between the noon of 
the 20th and 21st. 

Both longitude and right ascension are reckoned from the 
equinoctial point qp : longitude along the line °p JV ( which 
line is called the ecliptic), and right ascension along the 
celestial equator HP Q. 

Computing the length of the line °p JV, we find it equal to 
30' 36". 6 ; which was the sun's longitude at the instant of 
apparent noon, at Greenwich, March 21st, 1846. 
Latitude, Meridians of right ascension are at right angles to the celestial 
in astrono- eC | Ua t r ( at right angles to T $)• ^ ne fi rsi meridian runs 
what line through the point <¥>. Meridians of latitude are at right 
reckoned. angles to the ecliptic (at right angles to the line S JV). La- 
titude, in astronomy, is reckoned north and south of the ecliptic. 

Thus a star at m (Fig. 5), °p n, would he its longitude, n m 
its north latitude ; °p o its right ascension ; and o m its north 
declination. 
Path of the (44.) Thus, it may be perceived, that these observations 
snn# are very fruitful in giving important results ; but, as yet, we 

have used only two of them — those made on the 20th and 21st. 
By bringing the other observations into computation, and 
extending Fig. 5, we can find the points where the sun was 
on the other clays mentioned ; and then, by taking observa- 
tions every day in the year, the sun's right ascension and lon- 
gitude can be determined for every day : and its exact path- 
Len^-th of w °y through the apparent celestial sphere. The same kind 
a year, how f observations taken on the 20th, 21st, 22d, 23d, and 24th 
days of September, will show when the sun crosses the equa- 
tor from north to south; and how long it remains north of the 
equator; and how long south of it. In March, 1847, the 
same observations might have been made; and the exact 
length of an equinoctial year determined : and in this way that 
important interval has been decided, even to seconds. 

The true length of an equinoctial year was early a very 



SOLAR YEAR 37 

interesting problem to astronomers; and, "before they had Chap, hi. 
good clocks and, refined instruments, it was one of some diffi- 
culty to settle. But the more the difficulty, the greater the 
zeal and perseverance ; and we are often astonished at the 
accuracy which the ancients attained. 
The length of the equinoctial year, as stated in the tables of 

Days, hours, min. sees. 

Ptolomee, is - - - - 365 5 55 12 

Tycho Brahe, made it - 365 5 48 45 

Kepler, in his tables, - - - 365 5 48 57 

M. Cassini, in his tables, - - 365 5 48 52 

M. Be Lalande, - - - - 365 5 48 45 

Sir John Herschel, - - - 365 5 48 49.7 

The last cannot differ from the truth more than one or two solar and 

seconds. Let the reader notice that this is the equinoctial s,dereal 

year, 

year — the one that must ever regulate the change of sea- 
sons. There is another year — the sidereal year — which is 
about 20 minutes longer than the equinoctial year. The side- 
real year, is the time elapsed, from the departure of the sun 
from the meridian of any star, until it arrives at the same 
meridian again, and consists of 365 d. 6 A. 9 m. 9 5. 

As the stars are really the fixed points in space, this latter Cause of 
period is the apparent revolution of the sun ; and the shorter dlfferenC8, 
period, for the equinoctial year, is caused by the motion of 
the equinoctial points to the westward, called the precession 
of the equinoxes. Since astronomers first began to record 
observations, the fixed stars have increased, in right ascension, 
about 2 hours, in time, or 30 degrees of arc. 

The mean annual precession of the equinoxes is 50 ".1 of 
arc ; which will make a revolution, among the stars, in 25868 
years.* 

* The computation is thus : As 50".l is to the number of seconds in 
360 degrees ; so is one year to the number of years. Which gives 
25868 years, nearly. 

"We say, the stars increase in right ascension ; and this is true ; but 
the stars do not move — they are fixed; the meridian moves from the 
stars, 



D 



88 ASTRONOMY. 

CHAPTER IV. 

GEOGRAPHY OF THE HEAVENS. 

chap, iv . ^ 45, ) The fixed stars are the only landmarks in astrono- 
Groups of my, in respect to both time and space. They seem to have 
been thrown about in irregular and ill- defined groups and 
clusters, called constellations. The individuals of these groups 
and clusters differ greatly as to brightness, hue, and color ; 
but they all agree in one attribute — a high degree of perma- 
nence, as to their relative positions in the group; and the 
groups are as permanent in respect to each other. This has 
procured them the title of fixed stars ; an expression which 
must be understood in a comparative, and not in an absolute, 
sense ; for, after long investigation, it is ascertained that 
some of them, if not all, are in motion ; although too slow to 
be perceptible, except by very delicate observations, conti- 
nued through a long series of years. 
Magni- The stars are also divided into different classes, according 
stars 8 e ^° ^ ne ^ r d e g ree of brilliancy, called magnitudes. There are 
six magnitudes, visible to the naked eye; and ten telescopic 
magnitudes — in all, sixteen. 

The brightest are said to be of the first magnitude ; those 
less bright, of the second magnitude, etc. ; the sixth magni- 
tude is just visible to the naked eye. 
One star The stars are very unequally distributed among these 
of the first classes ; nor do all astronomers agree as to the number be- 
magm u e. j on gj n g £ eac |j . f or j^ j g impossible to tell where one class 
ends, and another begins ; nor is it important, for all this is 
but a matter of fancy, involving no principle. In the first 
magnitude there is really but one star ( Sirius ) ; for this is 
manifestly brighter than any other; but most astronomers 
put 15 or 20 into this class. 

The second magnitude includes from 50 to 60 ; the third, 
about 200, the numbers increasing very rapidly, as we descend 
in the scale of brightness. 

From some experiments on the intensity of light, it has 



GEOGRAPHY OF THE HEAVENS. 39 

been determined, that if we put the light of a star, of the Chap. iv. 
average 1st magnitude, 100, we shall have : 

1st magnitude = 100 4th magnitude = 6 
2d " = 25 5th " =2 

3d " = 12 6th " =1 

On this scale, Sir William Hersehel placed the brightness of 
Sirius at 320. 

Ancient astronomy has come down to us much tarnished 
with superstition, and heathen mythology. Every constella- 
tion bears the name of some pagan deity, and is associated 
with some absurd and ridiculous fable, yet, strange as it may 
appear, these masses of rubbish and ignorance — these clouds 
and fogs, intercepting the true light of knowledge, are still 
not only retained, but cherished, in many modern works, and 
dignified with the name of astronomy. 

Merely as names, either to constellations or to individual Ancient 
stars, we shall make no objections; and it would be useless, f ames must 

. . be conti- 

if we did ; for names long known, will be retained, however RUe d. 
improper or objectionable ; hence, when we speak of Orion, 
the Little Dog, or the Great Bear, it must not be understood 
that we have any great respect for mythology. 

It is not our purpose now to describe the starry heavens — 
to point out the variable, double, and multiple stars — the 
Milky Way and nebulae ; these will receive special attention 
in some future chapter ; at present, our only aim is to point 
out the method of obtaining a knowledge of the mere ap- 
pearance of the sky, to the common observer, which may be 
called the geography of the heavens. 

To give a person an idea of locality, on the earth, we refer 
to points and places supposed to be known. Thus, when we 
say that a certain town is 15 miles north-west of Boston, a 
ship is 100 miles east of the Cape of Good Hope, or a cer- 
tain mountain 10 miles north of Calcutta, we have a pretty 
definite idea of the localities of the town, the ship, and the 
mountain, on the face of the earth, provided we have a clear 
idea of the face of the earth, and know the position of Boston, 
the Cape of Good Hope, and Calcutta. 

So it is with the geography of the heavens ; the apparent 
4 



40 ASTRONOMY. 

Chap, iv . surface of the whole heavens must be in the mind, and then 
the localities of certain bright stars must be known, as land- 
marks, like Boston, the Cape of Good Hope, and Calcutta. 
stars about Tff e g^aH now ma ke some effort to point out these land- 
marks. The North Star is the first, and most important to 
be recognized ; and it can always be known to an observer, in 
any northern latitude, from its stationary appearance and alti- 
tude, equal to the latitude of the observer. At the distance of 
about 32 degrees from the pole, are seven bright stars, between 
the 1st and 2d magnitudes, forming a figure resembling a 
dipper, four of them forming the cup, and three the handle. The 
two forming the sides of the cup, opposite to the handle, are 
always in a line with the North Star ; and are therefore called 
pointers ; they always point to the North Star. The line join- 
ing the equinoxes, or the first meridian of right ascension, 
runs from the pole, between the other two stars forming the 
cup. The first star in the handle, nearest the cup, is called 
Alioth, the next Mizar, near which is a small star, of the 4th 
magnitude; the last one is Benetnasch. The stars in the 
handle are said to be in the tail of the Great Bear. 

About four degrees from the pole star, is a star of the 3d 
magnitude, e TJrsw Minoris. A line drawn through the pole 
(not pole star), and this star will pass through, or very near, 
the poles of the ecliptic and the tropics. A small constella- 
tion, near the pole, is called Ursa Minor, or the Little Bear. 
An irregular semicircle of bright stars, between the dipper 
and the pole, is called the Serpent. 
imaginary jf a \{ ne ]j e drawn from i Ursce Minoris, through the pole 
staAo st r a ° r m star, and continued about 45 degrees, it will strike a very 
beautiful star, of the 1st magnitude, called Capella. Within 
five degrees of Capella are three stars, of about the 4th mag- 
nitude, forming a very exact isosceles triangle, the vertical 
angle about 28 degrees. A line drawn from Alioth, through 
the pole star, and continued about the same distance on the 
other side, passes through a cluster of stars called Cassiopea 
in Iter chair. The principal star in Cassiopea, with the pole 
star and Capella, form an isosceles triangle, Capella at the 
vertex. 



GEOGRAPHY OF THE HEAVENS. 41 

( 46. ) More attention has been paid to the constellations Chap, iv. 
along the equator and ecliptic, than to others in remoter Ecliptic 
regions of the heavens, because the sun, moon, and planets, defined - 
traverse through them. The ecliptic is the sun's apparent 
annual path among the stars ( so called because all eclipses, 
of both sun and moon, can take place only when the moon is 
either in or near this line). 

Eight degrees on each side of the ecliptic is called the Signs of 
zodiac ; and this space the ancients divided into 12 equal x ie z0 iac ' 
parts ( to correspond with the 12 months of the year ), and 
each part (30°) is called a sign — and the whole, the 
signs of the zodiac. These divisions are useless ; and, of late 
years, astronomers have laid them aside; yet custom and 
superstition will long demand a place for them in the common 
almanacs. 

The signs of the zodiac, with their symbolic characters, are 
as follows: Aries T, Taurus & , Gemini n, Cancer vq, Leo ££, 
Virgo n}7, Libra ^, Scorpio rf|, Sagittarius £ , Capricornus YJ, 
Aquarius ox, Pisces X- 

Owing to the precession of the equinoxes, these signs do 
not correspond with the constellations, as originally placed ; 
the variation is now about 30 degrees; the stars remain in 
their places; and the first meridian, or first point of Aries, 
has drawn back, which has given to the stars the appearance 
of moving forward. 

Beginning with the first point of Aries as it now stands, Method of 
no prominent star is near it ; and, going along the ecliptic to tra <>ing the 
the eastward, there is nothing to arrest special attention, 
until we come to the Pleiades, or Seven Stars, though only 
six are visible to the naked eye. This little cluster is so well 
known, and so remarkable, that it needs no description. South- 
east of the Seven Stains, at the distance of about 18 degrees, 
is a remarkable cluster of stars, said to be in the BulVs Head; 
the largest star, in this cluster, is of the 1st magnitude, of a 
red color, called Aldebaran. It is one of the eight stars se- 
lected as points from which to compute the moon's distance, 
for the assistance of navigators. 

This cluster resembles an A, when east of the meridian, and 



stars. 



42 ASTRONOMY. 

Chap, iv . a y, when west of it. The Seven Stars, Aldeharan, and Ca- 
pella, form a triangle very nearly isosceles — Capella at trie 
vertex. A line drawn from the Seven Stars, a little to the 
west of Aldeharan, will strike the most remarkable constella- 
tion in the heavens, Orion (it is out of the zodiac, however,) ; 
some call it the Ell and Yard. The figure is mainly distin- 
guished by three stars, in one direction, within two degrees 
of each other ; and two other stars, forming, with one of the 
three first mentioned, another line, at right angles with the 
first line. 

The five stars, thus in lines, are of the 1st or 2d magnitude. 
A line from the Seven Stars, passing near Aldeharan and 
through Orion, will pass very near to Sirius, the most bril- 
liant star in the heavens. The ecliptic passes about midway 
between the Seven Stars and Aldebaran, in nearly an eastern 
direction. Nearly due east from the northernmost and bright- 
est star in Orion, and at the distance of about 25 degrees, is 
the star Procyon ; a bright, lone star. 

The northernmost star in Orion, with Sirius and Procyon, 
form an equilateral triangle. 
The con- Directly north of Procyon, at the distances of 25 and 30 
steiiations <w re es, are two bright stars, Castor and Pollux. Castor is 

are above the ° ° 

horizon, and the most northern. Pollux is one of the eight lunar stars. 
visible every Thus we might run over that portion of the heavens which is 
ring the win- ever visible to us ; and by this method every student of astro- 
ter season, nomy can render himself familiar with the aspect of the sky ; 
but it is not sufficiently definite and scientific to satisfy a ma- 
thematical mind. 

(47.) The only scientific method of defining the position 
of a place on the earth, is to mention its latitude and longitude ; 
and this method fully defines any and every place, however 
unimportant and unfrequented it may be : so in astronomy, the 
only scientific methods of defining the position of a star, is to 
mention its latitude and longitude, or, more conveniently, its 
, right ascension and declination. 

General J 

and indefi- It is not sufficient to tell the navigator that a coast makes 
mte descnp- Q ^ « n svich a direction from a certain point ; and that it is so 

tions not sa- x 

tisfactory. far to a certain cape ; and, from one cape to another, it is 



GEOGRAPHY OF THE HEAVENS. 43 

about 40 miles south-west — he would place very little reli- Chap, iv. 
ance on any such directions. To secure his respect, and what con- 
command his confidence, the latitude and longitude of every scutes a de- 
point, promontory, river, and harbor, along the coast, must be scriptioru 
given; and then he can shape his course to any point, or 
strike in upon it from the indefinite expanse of a pathless sea. 
So with an astronomer; while he understands and appreciates 
the rough and general descriptions, such as we have just given, 
he requires the certain description, comprised in right ascension 
and declination. 

Accordingly, astronomers have given the right ascensions 
and declinations of every visible star in the heavens ( and of 
very many that are invisible ), and arranged them in tables, 
in the order of right ascension. 

There are far too many stars, for each to have a proper John Bay- 
name; and, for the sake of reference, Mr. John Bayer, of er * " ie ° 

'■ ■'of reference- 

Augsburg, in Suabia, about the year 1603, proposed to denote 

the stars by the letters of the Greek and Roman alphabets ; 

by placing the first Greek letter, «, to the principal star in 

the constellation; $ to the second in magnitude; y to the 

third; and so on; and if the Greek alphabet shall become 

exhausted, then begin with the Roman, a, b, c, etc. 

" Catalogues of particular stars, in sections of the heavens, Particular 
have been published by different astronomers, each author catalo s ues - 
numbering the individual stars embraced in his list, according 
to the places they respectively occupy in the catalogue." 
These references to particular catalogues are sometimes 
marked on celestial globes, thus ; 79 H ; meaning that the 
star is the 79th in Herschel's catalogue; 37 M, signifies the 
37th number in the catalogue of Mayer, etc. 

Among our tables will be found a catalogue of a hundred 
of the principal stars, inserted for the purpose of teaching a defi- 
nite and scientific method of making a learner acquainted with the 
geography of the heavens. 

To have a clear understanding of the method we are about 
to explain, we again consider that right ascension is reckoned 
from the equinox, eastward along the equator, from h. to 
24 hours. When the sun comes to the equator, in March, its 



44 ASTRONOMY. 

Chap. rv. right ascension is ; and from that time its right ascension 
increases about four minutes in a day, throughout the year, 
to 24 hours : and then it is again at the equinox, and the 24 
hours are dropped. 
when it is -^ u * whatever be the right ascension of the sun, it is appa- 
apparent rent noon when it comes to the meridian ; and the more east- 
ward a body is, the later it is in coming to the meridian. Thus, 
if a star comes to the meridian at two o'clock in the afternoon 
( apparent time ), it is because its right ascension is two hours 
greater than the right ascension of the sun. 

Therefore, if from the right ascension of a star we subtract 

the right ascension of the sun, the remainder will be the time 

for that star to come to the meridian. 

Connection -^ we P ut ( -# * ) to represent the star's right ascension; 

between R, and (R O) to represent that of the sun; and T to represent 

ridian pas- ^ ie a PP aren l ^ me that the star passes the meridian, then we 

sage shall have the following equation : 

By transposition . . JR$cz=BO-\~T; 

That is, the right ascension of a star ( or any celestial body ), is 
equal to the right ascension of the sun, increased by the time that 
the star ( or body ) comes to the ■meridian. 

The right ascension of the sun is given, in the Nautical 
Almanac ( and in many other almanacs ), for every day in the 
year, when the sun is on the meridian of Greenwich; but 
many of the readers of this work may not have such an alma- 
nac at hand, and, for their benefit, we give the right ascen- 
sion for every fifth day of the year 1846 ( Table III); the 
local time is the apparent noon at Greenwich. 

We take the year 1846, because it is the second year after 
leap year; and the sun's right ascension for any day in that 
year, will not differ more than two minutes from its right 
ascension, on the same day, of any other year ; and will cor- 
respond with the right ascension of the same day in 1850, by 
adding 7f^ seconds ; and so on for each succeeding period 
of four years. 

To apply the preceding equation, the observer should ad- 
just his watch to apparent time ; that is, apply the equation 



GEOGRAPHY OF THE HEAVENS. 45 

of time, and know the direction of Ms meridian, at least chap. iv. 
approximately. In short, by the range of definite objects, 
he must be able to decide, within two or three minutes, when a 
celestial body is on his meridian. 

Thus, all prepared, we will give a few 

EXAMPLES. 

1. On the 20th of May ( no matter what year, if not many Examples 
years from 1850), in the latitude of 40° J\\ and longitude of tofind stars - 
80° W., at 9 h. 24 m. in the evening, clock time, I observed a 
lone, bright star, of about the 2d magnitude, on the meridian. It 
had a bland, white light ; and, as I had no instrument to mea- 
sure its altitude, I simply judged it to be 4:2°, What star 
was it ? 

We decide the question thus : 

Time per watch, - - - 9 h. 24 m. 00 s. 
Equation of time ( see Table ), add 3 46 

Apparent time, 9 27 46 

Lon. 80° W., equal, in time, to 5 20 00 

Apparent time, at Greenwich, - 14 47 46 

The right ascension of the sun, on the 20th of May (noon, Correction 

Greenwich time), is 3h. 47m. 15 s. ( see Table III). The 

... R. a 
increase, estimated at the rate of 4 minutes in 24 hours, will 

give 1 minute in 6 hours, or 10 seconds to 1 hour ; this, for 

14 h. 47 m., gives 2 m. 27 s. 

Hence, the right ascension of the sun, at the time of obser- 
vation, was - - - - 3 h. 49 m. 42 s. 
Apparent time of observation, - 9 27 46 
Eight ascension of the star, - - 13 h. 17 m. 28 s. 

By inspecting the catalogue of the stars ( Table II ), we 
find the right ascension of Spica to be 13 h. 17 m. 08 s., and its 
declination, 10° 21' 35". 

But, in the latitude of 40° N., the meridian altitude of the 
celestial equator must be 50° ; and any stars south of that 
must be of a less altitude. Therefore, the meridian altitude 
of Spica must be 50°, less 10° 21', or 39° 39' ; but the star 
f observed, I simply judged to have had an altitude of 42°. 



of the sun' 



46 ASTRONOMY. 

Chap. iv. It is very possible that I should err, in altitude, two or three 

degrees ; * but, it is not possible that the star I observed should 

be any other star than Spica ; for there is no other bright star 

near it. This is one of the lunar stars. 

Personal Being now certain that this star is Spica, I can observe it 

observations j n re lation to its appearance — the small stars that are near 

recommend- . ■ . . 

€d# it, and the clusters of stars that are about it — or the fact, 

that no remarkable constellation is near it. In short, I can 
so make its acquaintance as to know it ever after ; but I am 
unable to convey such acquaintance to others, by language ; 
true knowledge, in this particular, demands personal obser- 
vation, 
continua- % Qn tU u , f July, 1846, at 9 k. 34m., P. M., mean 

turn of exam- , y J *' ' ' ' p 

pies to find time per watch, a star of the 1st magnitude came to the meridian. 
stars. I was in latitude 39° JV., and about 75° W. The star was of 

a deep red color, and, as near as my judgment could decide, its 
altitude was between 25° and 30°. Two small stars were near 
it, and a remarkable cluster of smaller stars were west and north- 
west of it, at the distances of 5°, 6°, or 7°. What star was this ? 
Time per watch, - - - - 9 h. 34 m. 00 s. 
Equa. of time ( subtr. from mean time ) 3 48 

Apparent time, - - - - 9 30 12 

Longitude, 75°, equal to - - 5 

Apparent time, at Greenwich, - - 14 h. 30 m. 00 s. 
By examining the table for the sun's R. A., I find that, 
On the 1st of July, it is - - 6h. 40 m. 00 s. 
On the 5th, - - - - 6 56 30 

Variation, for 4 days, - - - 16 m. 30 s. 

At this rate, the variation for 2 days, 14^ hours, cannot be 

* Ten or twenty degrees, near the horizon, is apparently a much 
larger space than the same number of degrees near the zenith. Two 
stars, when near the horizon, appear to be at a greater distance asunder 
than when their altitudes are greater. The variation is a mere optical 
illusion; for, by applying instruments, to measure the angie in the 
different situations, we find it the same. Unless this fact is taken into 
consideration, an observer will always conceive the altitude of any ob- 
ject to be greater than it really is, especially if the altitude is less thaa 
45 degrees. 



GEOGRAPHY OF THE HEAVENS. 47 

far from 10 m. .10 s. ; and the right ascension of the sun, at Chap rv 

the time of observation, must have been An exam- 

Nearly ..-_-- 6K. 50 m. 10 B , •££*■« 

To which add, apparent time, - - 9 30 12 
Right ascension of the star, - - 16 h. 20 m. 22 s. 
By inspecting the catalogue of stars, I find Antares to have 

a right ascension of 16h. 20m. 2s. and a declination of 26° 4', 

south. 

In the latitude mentioned, the meridian altitude of the 

celestial equator must be 50° 0' 

Objects south of that plane must be less, hence (sub.) 26 4 

Meridian altitude of Antares, in lat. 50°, 23° 56 

As the observation corresponds to the right ascension of An- 
tares ( as near as possible, considering errors in observation, 
and probably in the watch ), and as the altitudes do not 
differ many degrees ( within the limits of guess work ), it is 
certain that the star observed was Antares. By its peculiar 
red color, and the remarkable clusters of stars surrounding it, 
I shall be able to recognize this star again, without the 
trouble of direct observation. 

3. On the night of the 20th of June, 1846, latitude 40° iV!, and To find 
longitude 75° W., at 1 h. 48 m. past midnight, clock time, lob- Altair - 
served a star of the 1st magnitude nearly on the meridian; two 
other stars, of about the 3d magnitude, within 3° of it ; the three 
stars forming nearly a right line, north and south ; the altitude 
of the principal star about 60°. What star was it? 

In these examples, the time must be reckoned on from noon 
to noon again ; therefore 1 h. 48 m. after midnight must be 
written, - 

Equation of time, to subtract, - 

Apparent time, - 
Longitude, - 

Greenwich apparent time, June 20, 

Sun's right ascension, at this time, 
Time, - 

Star's right ascension, - - 19 h. 44 m. 28 s. 



13 h. 


48 m. 


00 s. 




1 


12 


13 


46 


48 


5 






18 h. 


46 m. 


48 s. 


5h. 


57 m. 


40 s. 


13 


46 


48 



4b ASTRONOMY. 

Chap. iv. By inspecting the catalogue of stars, we find the right 
ascension of Altair 19 h. 43 m. 15 s.., and its declination 8° 
27' M; In latitude 40° N., the declination of 8° 27' N. will 
give a meridian altitude of 58° 27' ; and, in short, I know 
the star observed must be Altair, and the two other stars, 
near it, I recognize in the catalogue. 

By taking these observations, any person may become ac- 
quainted with all the principal stars, and the general aspect 
of the heavens ; but no efforts, confined merely to the study 
of books, will accomplish this end. 

The equation in Art. 47 is not confined to a star ; it may 
be any heavenly body, moon, comet, or planet. The time of 
passing the meridian is but another term for right ascension. 
If observations are made on any bright star, and no corre- 
sponding star is found in the catalogue, such a star would 
probably be a planet; and if a planet, its right ascension 
will change. 
The South- /-^ n rpk e ^qjq region of stars south of declination 50°, 

em Cross, . V J . & ' 

and Magei- is never seen m latitude 40° north, nor from any place nortn 
lan Clouds. f that parallel ; and, to register these stars in a catalogue, it 
has been necessary for astronomers to visit the southern 
hemisphere, as we have before mentioned ; but these stars 
are mostly excluded from our catalogues. There are several 
constellations, in the southern region, worthy of notice — the 
Southern Cross and the Magellan Clouds. The Southern 
Cross very much resembles a cross ; so much so, that any 
person would give the constellation that appellation. Its 
principal star is, in right ascension, 12 h. 20 m., and south 
declination 33°. 

The Magellan Clouds were at first supposed to be clouds 
by the navigator Magellan ; who first observed them. They 
are four, in number ; two are white, like the Milky Way, and 
have just the appearance of little white clouds. They are 
nefadce. The other two are black — extremely so — and are 
supposed to be places entirely devoid of all stars ; yet they 
are in a very bright part of the Milky Way : Bight ascen- 
sion, 10 h. 40 m., decimation, 62° south. 



DESCRIPTIVE ASTRONOMY. 



49 



SECTION II. 



DESCRIPTIVE ASTRONOMY. 



CHAPTER I. 



FIKST CONSIDERATIONS AS TO THE DISTANCES OF THE HEAVENLY 
BODIES. SIZE AND EXACT FIGURE OF THE EARTH. 



( 49.) Hitherto we have con- 
sidered only appearances, and 
have not made the least inquiry, 
as to the nature, magnitude, or 
distances of the celestial objects. 

Abstractly, there is no such 
thing as great and small, near 
and remote; relatively speaking, 
however, we may apply the terms 
great, and very great, as regards 
both magnitude and distance. 
Thus an error of Un feet, in the 
measure of the length of a 
building, is very great — when 
an error of ten rods, in the mea- 
sure of one hundred miles, would 
be too trifling to mention. 

Now if we consider the dis- 
tance to the stars, it must be 
relative to some measure taken 
as a standard, or our inquiries 
will not be definite, or even in- 
telligible. We now make this C 
general inquiry : Are the heavenly bodies near to, or remote from, 
the earth? Here, the earth itself seems to be the natural 
standard for measure ; and if any body were but two, three, 
or even ten times the diameter of the earth, in distance, we 
4 E 




Chap. I. 

Distance 
is but rela- 
tive. 



Are the 
heaven] j' bo- 
dies remote 1 



50 ASTRONOMY. 

cha^i. should call it near; if 100, 200, or 2000 times the diametei 
of the earth, we should call it remote. To answer the 
inquiry. Are the heavenly bodies near or remote ? we must put 
them to all possible mathematical tests ; a mere opinion is of 
no value, without the foundation of some positive knowledge. 
Let 1, 2 ( Fig. 6 ), represent the absolute position of two 
stars ; and then, if A B C represents the circumference of the 
earth, these stars may be said to be near ; but if a b c repre- 
sents the circumference of the earth, the stars are many times 
the diameter of the earth, in distance, and therefore may 
The means he said to be remote. If AB C is the circumference of 
this question * ne ear * n j £ n relation to these stars, the apparent distance of 
pointed out. the two stars asunder, as seen from A, is measueed by the 
angle 1 A 2 ; and their apparent distance asunder, as seen 
from the point B, is measured by the angle 1 B 2 ; and when 
the circumference AB C is very large, as represented in our 
figure, the angle A, between the two stars, is manifestly 
greater than B. But if ah c is the circumference of the 
earth, the points a and 6 are relatively the same as A and B. 
And, it is an ocular demonstration that the angle under which 
the two stars would appear at a, is the same, or nearly the 
same, as that under which they would appear at h ; or, at 
least, we can conceive the earth so small, in relation to the 
distance to the stars, that the angle under which two stars 
would appear, would be the same seen from any point on the 
earth. 
The con- Conversely, then, if the angle under which two stars appear 
is the same as seen from all parts of the earth's surface, it is 
certain that the diameter of the earth is very small, compared 
with the distance to the stars ; or, which is the same thing, 
the distance to the stars is many times the diameter of the earth. 
Therefore observation has long since decided this important 
point. Sir John Herschel says : " The nicest measurements 
of the apparent angular distance of any two stars, inter se, 
taken in any parts of their diurnal course ( after allowing for 
the unequal effects of refraction, or when taken at such times 
that this cause of distortion shall act equally on both ), mani- 
fest not the slightest perceptible variation. Not only this, but 



elusion. 



COMPARATIVE DISTANCES. 



51 




Another 
illustration 
of the great 
distance to 
the stars. 



at whatever point of the earth's surface the measurement is chap^i. 
performed, the results are absolutely identical. No instruments 
ever yet invented by man are delicate enough to indicate, by 
an increase or diminution of the angle subtended, that one 
point of the earth is nearer to or farther from the stars than 
another." 

( 50.) Perhaps the following view of this subject will be 
more intelligible to the general reader. 

Let Z HN 
II represent 
the celestial 
equator, as 
seen from the 
equator on 
the earth; and 
if the earth be 
large, in rela- 
tion to the 
distance to 
the stars, the 
observer, will 
be at z' ; and 
the part of the 

celestial arc above his horizon, would be represented by AZ B, 
and the part below his horizon by A NB, and these arcs are ob- 
viously unequal ; and their relation would be measured by the 
time a star or heavenly body remains above the horizon, com- 
pared with the time below it ; but by observation ( refraction 
being allowed for ), we know that the stars are as long above 
the horizon as they are below; which shows that the ob- 
server is not at z r , but at z, and even more near the center ; 
so that the arc A Z B, is imperceptibly unequal to the arc H 
N H\ that is, they are equal to each other; and the earth 
is comparatively but a point, in relation to the distance to 
the stars. 

This fact is well established, as applied to the fixed stars, 
sun. and planets ; but with the moon it is different ; that body 



an 
tion 



The moon 
excep- 



52 



ASTRONOMY. 



Chap ' *• is longer below the liorizon than above it ; which shows that 
its distance from the earth is at least measurable. 

( 51.) It is improper, at present, or rather, it is too advanced 
an age, to pay any respect to the ancient notion, that the earth 
is an extended plane, bounded by an unknown space, inacces- 
sible to men. Common intelligence must convince even the 
child, that the earth must be a large ball, of a regular, or an 
irregular shape; for every one knows the fact, that the earth 
has been many times circumnavigated; which settles the 
question. 
Earth's In addition to this, any observer may convince himself, that 
surface con- the surface of the sea, or a lake, is not a plane, but everywhere 

vex„ 

convex ; for, in coming in from sea, the high land, back in the 
country, is seen before the shore, which is nearer the observer; 
the tops of trees, and the tops of towers, are seen before their 
bases. If the observer is on shore, viewing an approaching 
vessel, he sees the topmast first ; and from the top, downward, 
the vessel gradually comes in view. This being the case on 
every sea, and on every portion of the earth, proves that the 
surface of the earth is convex on every part — hence if must 
be a globe, or nearly a globe. These facts, last mentioned, 
are sufficiently illustrated by 

Fig. 8. 



(52.) On the supposition that the earth is a sphere, there 
are several methods of measuring it, without the labor of 
applying the measure to every part of it. The first, and 
most natural method (which we have already mentioned), is 
that of measuring any definite portion of the meridian, and 
from thence computing the value of the whole circumference. 
How to Thus, if we can know the number of degrees, and parts of 
find the cu- ft (Wree, in the arc AB (Fig. 9), and then measure the dis- 

cumference °"; " ■ vox 

of the earth, tance in miles, we in fact virtually know the whole circumfe- 



DIAMETER OF THE EARTH 



53 



AP. I. 



How to 
find the dia- 
meter. 



rence ; for whatever part the arc A B is of 360 degrees, the ch 
same part, the number of miles in A B, is of the miles in the 
whole circumference. 

To find the arc A B, the latitudes of the two points, A and 
B, must be very accurately taken, and their difference will 
give the arc in degrees, minutes, and seconds. Now A B must 
be measured simply in distance, as miles, yards, or feet; but 
this is a laborious operation, requiring great care and perse- 
verance, To measure directly any considerable portion of a 
meridian, is indeed impossible, for local obstructions would 
soon compel a deviation from any definite line ; but still the 
measure can be continued, by keeping an account of the de- 
viations, and reducing the measure to a meridian line. 

Let m be the miles or feet in A B ; then the whole circum- 
ference will be expressed by ( 

( 53. ) When we know the 
hight of a mountain, as re- 
presented in Fig. 9, and at 
the same time know the dis- 
tance of its visibility from 
the surface of the earth; 
that is, know the line M A ; 
then we can compute the 
line M C, by a simple theo- 
rem in geometry ;, thus, 

CMXMB=(AM) 2 ; 
n „- (AMY 

Now as the right hand 
member of this equation is known. C M is known; and as 
part of it ( MB ) is already known, the other part, B C, the 
diameter of the earth, thus becomes known. 

This method would be a very practical one, if it were not objection 
for the uncertainty and variable nature of refraction near the t0 this me - 
borizon ; and for this reason, this method is never relied upon, lhod 
although it often well agrees with other methods. As an ex- 
ample under this method, we give the following : 




54 ASTRONOMY. 

chap, i. A mountain, two miles in perpendicular hight, was seen 
from sea at a distance of 126 miles. If these data are cor- 
rect, what then is the diameter of the earth 

Solution; MC=^^- =63x126=7938. 5(7=7936. 

Dip of the ( 54. ) This same geometrical theorem serves to compute 
horizon. the dip of the horizon. The true horizon is a right angle from 
the zenith ; hut the navigator, in consequence of the motion 
of his vessel, can never use the true horizon ; he must use 
the sea offing, making allowance for its dip. If the naviga- 
tor's eye were on a level with the sea, and the sea perfectly 
stable, the true and apparent horizon would be the same. 
But the observer's eye must always be above the sea ; and 
the higher it is, the greater the dip ; and the amount of dip 
will depend on the hight of the eye, and the diameter of the 
earth. The difference between the angle AMC (Fig. 9), 
and a right angle ( which is the same as the angle A EM), 
is the measure of the dip corresponding to the hight B M. 

For the benefit of navigators, a table has been formed, 
showing the dip for all common elevations.* 



* The dip is computed thus : 

The angle 
at the center Put BC (Fig. 9) =JD, BM=h\ 

is equal to / T) 

the dip. Then EM= ( « -H) 5 and {MAY = CMxMB=(JD+h)h. 

By trigonometry, (EA) 2 : (MA)* : : R 2 : tfm*AEM; 

B 2 
That is, - - - — , : (B-\-h)h : : R 2 : tm. 2 AEM 

For very moderate elevations, h is extremely small, in rela- 
tion to D ; and the second term of the proportion may be 
Dh. (R represents the radius of the tables.) Making this 
consideration, we have 

B 2 

-j- : Dh : : R 2 : tsni. 2 AEM; 
4 

Or, - - I) : h : : 4R 2 : t&n. 2 AEM; 
Or, - - JD: Jl : : 2R : tm.AEM. 



DIP OF THE HORIZON. 55 

( 55. ) All such computations are made on the supposition Chap. i. 
that the earth is exactly spherical ; and it is, in fact, so nearly 
spherical, that no corrections are required in consequence of 
its deviation from that figure. 

After correct views began to he entertained, as to the mag- The earth 
nitude of the earth, and its revolution on an axis, philosophers not ex ^ctiy 
concluded that its equatorial diameter might be greater than " 
its polar diameter; and investigations have been made to 
decide the fact. 

If the earth were exactly spherical, it is plain that the cur- 
vature over its surface would be the same in every latitude; 
but if not of that figure, a degree would be longer on one part 
of the earth than on another, " But," says Herschel, "when 
we come to compare the measures of meridional arcs made in 
various parts of the globe, the results obtained, although they 
agree sufficiently to show that the supposition of a spherical 
figure is not very remote from the truth, yet exhibit discord- 
ances far greater than what we have shown to be attributable 
to error of observation ; and which render it evident that the 
hypothesis, in strictness of its wording, is untenable. The 
following table exhibits the lengths of a degree of the meri- 
dian ( astronomically determined as above described), ex- 



By inspecting this last proportion, it will be perceived that 
the tangent of the dip varies as the square root of the eleva- 
tion. To apply this proportion, we adduce the following 
problem : 

The diameter of the earth is 7912 miles ; the elevation of 
the eye, above the surface, is ten feet. What is the dip? 



2E . . log. 




.10.301030 


i/T, . log. 




.500000 


Product of the means (log.), - - - - 


10.801030 


2) miles, 7912, 


- - log. - 3.898286 




Feet, - 5280, 


- - log. - 3.722634 
2 ) 7.620920 




JD in feet, - 


- (log.) 3.810460 . 


. 3 810460 




tan. 3' 22"' - - - 


6.990570 



56 



ASTRONOMY. 



chap, i . pressed in British standard feet, as resulting from actual 
measurement, made with all possible care and precision, by 
commissioners of various nations, men of the first eminence, 
supplied by their respective governments with the best instru- 
ments, and furnished with every facility which could tend to 
insure a successful result of their important labors. 



Country. 



Sweden 

Russia 

England 

France 

France . . , 

E ome 

America, U. S.. . 
Cape of G. Hope 
India .......... 

India 

Peru 



Latitude 

of Middle of 

the Arc. 



66 20 10 
58 17 37 
52 35 45 
46 52 2 
44 51 2 
42 59 
39 12 
33 18 30 
16 8 22 
12 32 21 
1 31 



a, c !Lengthof| 

A c , Degree ] 

measured. < f . ■, 

concluded 



Observers. 



1°37' 
3 35 
3 57 
8 20 
12 22 

2 9 
I 28 
1 13 

15 57 
1 34 

3 7 



19" 

5 
13 



13 
47 
45 

m 

40 

56 

3 



365782 
365368 
364971 
364872 
364535 
364262 
363786 
364713 
363044 
363013 
362808 



Svanberg. 
Struve. 
Roy, Kater. 
Lacaille, Cassini. 
Delambre, Mechain. 
Boscovich, 
Mason, Dixon. 
Lacaille. 

Lambton, Everest. 
Lambton. 
Condamine, etc. 



The earth <•' It is evident, from a mere inspection of the second and 
-} -e oies f° ur * n columns of this table, that the measured length of a de- 
fchan at the gree increases with the latitude, being greatest near the poles, 

equator. ^ j eagt near ^ e q uator » 

"Assuming," continues Herschel, "that the earth is an 
ellipse, the geometrical properties of that figure enable us to 
assign the proportion between the lengths of its axes which 
shall correspond to any proposed rate of variation in its cur- 
vature, as well as to fix upon their absolute lengths, corre- 
sponding to any assigned length of the degree in a given 
latitude. Without troubling the reader with the investiga- 
tion (which may be found in any work on the conic sections), 
it will be sufficient to state that the lengths, which agree on 
the whole best with the entire series of meridional arcs, which 
have been satisfactorily measured, are as follow : — 

Feet. Miles. 

Greater, or equatorial diam., =41,847,426=7925.648 
Lesser, or polar diam., - - =41,707,620=7899.170 

Difference of diameters, or ^ q one- 26 4^8 

polar compression, - - - 

The propcrtioa of the diameters is very nearly that of 



FORM OF THE EARTH 



57 



298 : 299, and their difference j^j of the greater, or a very Chap, l 
little greater than ^lo" 

( 56. ) The shape of the earth, thus ascertained by actual 
measurement, is just what theory would give to a body of 
water equal to our globe, and revolving on an axis in 24 
hours ; and this has caused many philosophers to suppose that 
the earth was formerly in a fluid state. 

If the earth were a sphere, a plumb line at any point on Expiana- 
its surface would tend directly toward the center of gravity tl ° n of radlus 

•* m ° "f of curvature. 

of the body; but the earth being an ellipsoid, or an oblate 
spheroid, and the plumb lines, being perpendicular to the sur- 
face at any point, do not tend to the center of gravity of the 
figure, but to points as represented in Fig. 10. 

The plumb line at H tends to 
F, yet the mathematical center, 
and center of gravity of the 
figure, is at E. So at I, the 
plumb line tends to the point G; 
and as the length of a degree at 
A, is to the length of a degree 
at H, so is IG to IIF. If, 
however, a passage were made 
through the earth, and a body let drop through it, the body 
would not pass from /to G; its first tendency at /would be 
toward the point G; but after it passed below the surface, at 
I, its tendency would be more and more toward the point E, 
the center of gravity ; but it would not pass exactly through 
that point, unless dropped from the point A, or the point C. 

( 57. ) If the earth were a perfect and stationary sphere, F 0rce f 
the force of gravity, on its surface, would be everywhere the g ravit y diffe - 
same ; but, it being neither stationary, nor a perfect sphere, rent rts of 
the force of gravity, on the different parts of its surface, must the earth ; 
be different. The points on its surface nearest its center of 
gravity, must have more attraction than other points more 
remote from the center of gravity ; and if those points which 
are more remote from the center of gravity have also a rotary 
motion, there will be a diminution of gravity on that account. 

Let A B (Fig. 10) represent the equatorial diameter of 




58 



ASTRONOMY. 



Chap » *• the earth, and CD the polar diameter; and it is obvious 
that E will be the center of gravity, of the whole figure, and 
Gravity di- ^at ^h e f orce f gravity at Q and D will be greater than at 
rotation. an y other points on the surface, because E C, or ED, are 
less than any other lines from the point E to the surface. 
The force of gravity will be greatest on the points C and D, 
also, because they are stationary : all other points are in a 
circular motion ; and circular motion has a tendency to depart 
from the center of motion, and, of course, to diminish gravity. 
The diminution of the earth's gravity by the rotation on its 
axis, amounts to its 2J9 part,* at the equator. By this frac- 



Compnta- 
tion tof the 
amount of 
diminution. 



Fig. 11 



* Let D be the equatorial diameter 
of the earth, F the versed sine of an arc, 
corresponding to the motion in a second 
of time, and c the chord, or arc ( for the 
chord and arc of so small a portion of the 
circumference will coincide, practically 
speaking). 

A portion of the earth's gravity, equal 

to F, is destroyed by the rotation of the earth, and we are 

now to compute its value. 

By proportional triangles, F : c : : c : D; 




Or 



F= 



~D 



(1) 



The value of c is found by dividing the whole circumference 
into as many equal parts as there are seconds in the time of 
revolution. But the time of revolution is 23 h. 56 m. 4 s., ~ 
86164 seconds. 

The whole circumference is 



Therefore, 



(3.1416)D; 
(3.1416)D 



(2) 



By this value of c, we have F=- 



(86164) 
(3.1416 )2j) 
: (86164) 2 ' 

The visible force of gravity, at the equator, is the distance 
a body will fall the first second of time, expressed in feet. 
Let us call this distance g. Now the part of gravity des- 



EFFECT OF FORM ON GRAVITY. 59 

tion, then, is the weight of the sea about the equator lightened, Chap. i. 
and thereby rendered susceptible of being supported at a 
higher level than at the poles, where no such counteracting 
force exists. 

c 2 
troyed by rotation, as we have just seen, is -= ; therefore the 

c 2 \ 
whole force of gravity is (<7-j~7w 

Our next inquiry is; what part of the whole is the part de- Ratio of the 

diminution 

stroyed? Or what part of (#+77) * s 7)? 

Which, by common arithmetic, is, 
c 2 
D c 2 1 



9+i 9D ^ 2+1 

From™ - D'- ( 86164 ) ,e * or ° - < 86164 > 2 ■ 
*rom(Z) D - J^gy °h ~ 2 - ^t m yj) > 

Hence. 

gJD_ (86164)^ (86164)2(16.07) 

~c^~~ (3.1416)^ ""(3.1416)2 (7925)(5280)' 

By the application of logarithms, we soon find the value of 
this expression to be 288.4. Therefore, gB — - 



V+l 289.4 
c 2 ' 

We may now inquire, how rapidly the earth must revolve 
on its axis, so that the whole of gravity would be destroyed 
on the equator. That is, so that F shall equal g. Equation 

C 2 

(1) then becomes, g=^, or c= t JgD. 

But as often as c is contained in the whole circumference, 
is the corresponding number of seconds in a revolution; that 
is, the time in seconds must correspond to the expression, 



£±m or, (3.1416)V£. 



'gD 



60 ASTRONOMY. 

Chap - !• ( 58. ) It is this centrifugal force itself that changed the 
shape of the earth, and made the equatorial diameter greater 
than the polar. Here, then, we have the same cause, exer- 
cising at once a direct and an indirect influence. The amount 
Rotation f the former ( as we may see by the note ) is easily calcu- 
and indirect ^e& ; that of the latter is far more difficult, and requires a 
effect on gra- knowledge of the integral calculus; "But it has been clearly 
treated by Newton, Maclaurin, Clairaut, and many other emi- 
nent geometers ; and the result of their investigations is to 
show, that owing to the elliptic form of the earth alone, and 
independently of the centrifugal force, its attraction ought to 
increase the weight of a body, in going from the equator to 
the pole, by nearly its T i ^ * n P art 5 which, together with the 
jig- th part, due from centrifugal force, make the whole quan- 
tity T i ¥ th part ; which corresponds with observations as 
deduced from the vibrations of pendulums." — See Natural 
Philosophy. 

(59.) The form of the earth 
g * ' is so nearly a sphere, that it is 

considered such, in geography, 
navigation, and in the general 
problems of astronomy. 

The average length of a de- 

and geogra- mm WM \ \ ° ° 

phicai miles. fff \ \ gree is 69i English miles ; and, 

as this number is fractional, and 

inconvenient, navigators have ta- 

P citly agreed to retain the ancient, 

rough estimate of sixty miles to a degree ; calling the mile a 

geographical mile. Therefore, the geographical mile is longer 

than the English mile. 

D, in feet, = (7925)(5280) ; g = 16.076. By the applica- 
tion of logarithms, we find this expression to be 5069 seconds, 
or 1 h. 24 m. 29 s. ; which is about 17 times the rapidity of 
its present rotation. 

In a subsequent portion of this work, we shall show how 
to arrive at this result by another principle, and through 
another operation. 



English 




CONVERGENCY OF MERIDIANS. 61 

As all meridians come together at the pule, it follows that Chap, t, 
a degree, between the meridians, will become less and less as 
we approach the pole; and it is an interesting problem to 
trace the law of decrease.* 

* This law of decrease will become apparent, by inspecting 
Fig. 12. Let Eq represent a degree, on the equator, and 
Eq C a sector on the plane of the equator, and of course EC 
is at right angles to the axis C P. Let D EI he any plane 
parallel to Eq C; then we shall have the following proportion : 
EC : DI : : EQ : BE. 

In trigonometry, E C is known as the radius of the sphere ; 
D /as the cosine of the latitude of the point D (the nume- 
rical values of sines and cosines, of all arcs, are given in trigo- 
nometrical tables) : therefore we have the following rule, to 
compute the length of a degree between two meridians, on 
any parallel of latitude. 

Rule. — As radius is to the cosine of the latitude ; so is the 
length of a degree, on the equator, to the length of a parallel de- 
gree in that latitude. 

Calling a degree, on the equator, 60 miles, what is the Example. 
length of a degree of longitude, in latitude 42° ? 

SOLUTION BY LOGARITHMS. 

As radius (see tables), - - - 10.000000 
Is to cosine 42° (see tables), - - - 9.871073 
So is 60 miles (log.), - 1.778151 

To 44-jSg-Vo miles > 1.649224 

At the latitude of 60°, the degree of longitude is 30 miles ; 
the diminution is very slow near the equator, and very rapid 
near the poles. 

In navigation, the DE's are the known quantities ob- To reduce 
tained by the estimations from the log line, etc. ; and the de P arture t0 
navigator wishes to convert them into longitude, or, what ° ngl " 
is the same thing, he wishes to find their values projected on 
the equator, and he states the proportion thus : 
DI : EC : : DE : EQ; 
That is ; as cosine of latitude is to radius ; so is departure to 
difference of longitude. 

F 



62 



ASTRONOMY. 



CHAPTER II 



PARALLAX, GENERAL AND HORIZONTAL. RELATION BETWEEN 

PARALLAX AND DISTANCE. REAL DIAMETER AND MAGNI- 
TUDE OF THE MOON. 

Chap. h. ( 60. ) Parallax is a subject of very great importance in 
astronomy ; it is the key to the measure of the planets — to 
their distances from the earth — and to the magnitude of the 
whole solar system. 
Parallax in Parallax is the difference in position, of any body, as seen 
from the center of the earth, and from its surface. 

When a body is in the zenith of any observer, to him it has 
no parallax; for he sees it in the same place in the heavens, 
as though he viewed it from the center of the earth. The 
greatest possible parallax that a body can have, takes place 
when the body is in the horizon of the observer ; and this 
parallax is called horizontal parallax. Hereafter, when we 
speak of the parallax of a body, horizontal parallax is to be 
understood, unless otherwise expressed. 

A clear and summary illustration of parallax in general, is 
given by Fig. 13. 

Horizontal Fig> 13 J^et Q ^q 

parallax. , 

"* ihe center of 
the earth, Z 
the observer, 
and P, or p, 
the position 
of a body. 
From the 
center of the 
earth, the 
body is seen 
in the direc- 
tion of the 
line CP, or Cp; from the observer at Z, it. is seen in the 




PARALLAX. 63 

direction of Z P, "or Zp\ and the difference in direction, of Chap. ii. 
these two lines, is parallax. When P is in the zenith, there 
is no parallax ; when P is in the horizon, the angle Z P C is 
then greatest, and is the horizontal parallax. 

We now perceive that the horizontal parallax of any body Relation 
is equal to the apparent semidiameter of the earth, as seen from between P a - 
the body. The greater the distance to the body, the less the distance. 
horizontal parallax ; and when the distance is so great that 
the semidiameter of the earth would appear only as a point, 
then the body has no parallax. Conversely, if we can detect 
no sensible parallax, we know that the body must be at a 
vast distance from the earth ; and the earth itself appear as 
a point from such a body, if, in fact, it were even visible. 

Trigonometry gives the relation between the angles and 
sides of every conceivable triangle; therefore we know all 
about the horizontal triangle Z CP, when we know C Z and 
the angles. Calling the horizontal parallax of any body p, 
and the radius of the earth r, and the distance of the body 
from the center of the earth x ( the radius of the table always 
R, or unity), then, by trigonometry, we have, 

R : x : : sin. p : r ; 

Therefore, - - - x=( -. Jr. 

\sm.p/ 

From this equation we have the following general rule, to 
find the distance to any celestial body : 

Kule. — Divide the radius of the tables by the sine of the i> u i e t0 
horizontal parallax. Multiply that quotient by the semidiameter find th e dis- 
of the earth, and tlie product will be the residt. tances to the 

This result will, of course, be in the same terms of linear ^ A - 

DOQies. 

measure as the semidiameter of the earth ; that is, if r is in 
feet, the result will be in feet ; if r is in miles, the result will 
be in miles, etc. : but, for astronomy, our terrestrial measures 
are too diminutive, to be convenient (not to say inappropri- 
ate) ; and, for this reason, it is customary to call the semidia- 
meter of the earth unity ; and then the distance of any body 
from the earth is simply the quotient arising from dividing 
the radius, by the sine of the horizontal parallax, pertaining to 



64 ASTRONOMY. 

Chap^ii. the body ; and it is obvious, that the less the parallax, the 
greater this quotient ; that is, the greater the distance to the 
body; and the difficulty, and the only difficulty, is to obtain 
the horizontal parallax. 
Horizontal ( 61.) The horizontal parallax cannot be directly observed, 
not be ob- ^y reason of the great amount and irregularity of horizontal 
served. refraction ; but if we can obtain a parallax at any considera- 
ble altitude, we can compute the horizontal parallax there- 
from.* 

The fixed stars have no sensible horizontal parallax, as we 
have frequently mentioned ; and the parallax of the sun is 
so small, that it cannot be directly observed ( see 40 ) ; the 
moon is the only celestial body that comes forward and pre- 
sents its parallax ; and from thence we know that the moon 
is the only body that is within a moderate distance of the 
earth. 

That the moon had a sensible parallax, was known to the 
earliest observers, even before mathematical instruments were 
at all refined; but, to decide upon its exact amount, and 
detect its variations, required the combined knowledge and 
observations of modern astronomers. 

Deduction * I n the two triangles Zp C and ZP C (Fig. 13), call the 
angle p the parallax in altitude, and the angle ZP C = x, 
and Cp and CP each equal D. Then, by trigonometry, 
we have 

sin. pZC : sin.p :: D : r; 

And - - R : sin. x : : D : r. 
Therefore, by equality of ratios (see algebra), 

sin. pZC : sm.p : : R : sin. x. 

But the sine pZC is the cosine of the apparent zenith dis- 
tance. Therefore, 

R sin. p 

sin. x= : ; 

cos. zenith distance 

That is ; the sine of the horizontal parallax is equal to the sine 
of the parallax in altitude, into the radius, and divided by the 
cosine of the apparent zenith distance. 



:arallax. 



LUNAR PARALLAX. 65 

The lunar parallax was first recognized in European and chap. ii. 
northern countries, by its appearing to describe more than a By what 
semicircle south of the equator, and less than a semicircle north observatlons 

r . the lunar pa- 

of that line ; and, on an average, it was observed to be a longer ra u ax was 
time south, than north of the equator ; hut no such inequality first indica - 
could be ol served from the region of the equator. 

Observers at the south of the equator, observing the posi- 
tion of the moon, see it for a longer time north of the equator 
than south of it ; and, to them, it appears to describe more than 
a semicircle noi th of the equator. 

Here, then, we have observation against observation, unless 
we can reconcile them. But the only reconciliation that can 
be made, is to conclude that the moon is really as long in one 
hemisphere as the other ; and the observed discrepancy must 
arise from the positions of the observers ; and when we reflect 
that parallax must always depress the object ( see Fig. 13 ), 
and throw it farther from the observer, it is therefore per- 
fectly clear that a northern observer should see the moon 
farther to the south than it really is ; and a southern observer 
see the same body farther north than its true position. 

( 62.) To find the amount of the lunar parallax, requires 
the concurrence of two observers. They should be near the 
same meridian, and as far apart, in respect to latitude, as 
possible ; and every circumstance, that could affect the result, 
must be known. 

The two most favorable stations are Greenwich (England) Observa- 
and the Cape of Good Hope. They would be more favorable * l( !" s ^° ob ' 
if they were on the same meridian ; but the small change in mount of pa- 
declination, while the moon is passing from one meridian to ra ax 
the other, can be allowed for ; and thus the two observations 
are reduced to the same meridian, and equivalent to being 
made at the same time. 

The most favorable times for such observations, are when 
the moon is near her greatest declinations, for then the change 
of declination is extremely slow. 

Let A ( Fig. 14 ) represent the place of the Greenwich ob- 
servatory, and B the station at the Cape of Good Hope. 
C is the center of the earth, and Z and Z' are the zenith 



66 



ASTRONOMY. 



Chap. II. 



Fiff. 14. 



Illustration 
of primary 
observations. 




and finally, i/C* Now 



points of the observers. Let M 
be the position of the moon, and 
the observer at A will see it pro- 
jected on the sky at m', and the 
observer at B will see it pro- 
jected on the sky at m. 

Now the figure A C BM is a 
quadrilateral; the angle A CB 
is known by the latitudes of the 
two observers; the angles MA 
C and MB C are the respective 
zenith distances, taken from 180°. 

But the sum of all the angles 
of any quadrilateral is equal to 
four right angles ; and hence the 
angles at A, C, and B, being 
known, the parallactic angle at 
M is known. 

In this- quadrilateral, then, we 
have two sides, A C and CB, 
and all the angles ; and this is 
sufficient for the most ordinary 
mathematician to decide every 
particular in connection with it: 
that is, we can find AM, MB, 
MC being known, the horizontal 



A mathe- 
matical de- 
duction. 



* The direct and analytical method of obtaining MC, will be 
very acceptable to the young mathematician; and, for that 
reason, we give it. 

Put AC=CB=r, CM=x, and the two parts of the ob- 
served parallactic angle, M, represented by P and Q, as in 
the figure. Also, let a represent the natural sine of the angle 
M A C, and b the natural sine of the angle MB C : 

Then, by trigonometry, - x : a : : r : sin. Q ; 

Also, - x : b : : r : sin. P; 

Hence, - - - - sin. P+sin. Q—±-~±-'-. . 



x 



a) 



LUNAR PARALLAX. 67 

parallax can be computed, for it is but a function of the dis- Cha?. if. 
tance (see 60). 

By the equation (Art. 60), x=l )r 

By changing, - - - sin. p=( )r; and when x, the 

distance, is known, sin. p, or sine of the horizontal parallax, 
is known. 

( 63. ) The result of such observations, taken at different Variable 
times, show all values to M C, between 55 T 9 /o, and 63 T Vo ; distance t0 

' ' 100 ' * ° ° ' the moon. 

taking the value of r as unity. 

These variations are regular and systematic, both as to 
time and place, in the heavens ; and they show, without fur- 
ther investigation, that the moon does not go round the earth 
in a circle, or, if it does, the earth is not in the center of that 
circle. 

The parallaxes corresponding to these extreme distances, 
are 61' 29" and 53' 50". 

When the moon moves round to that part of her orbit Apogee 
which is most remote from the earth, it is said to be in apogee; P er 'g ee - 
and, when nearest to the earth, it is said to be in perigee. 
The points apogee and perigee, mainly opposite to each other, 
do not keep the same places in the heavens, but gradually 
move forward in the same direction as the motion of the moon, 
and perform a revolution in a little less than nine years. 



But, by a general theorem in trigonometry, 

sin. P+ sin. Q=2 sin.^i^, cos.— ^. . (2) 

A A 

Now by equating (1) and (2), and observing that P-\-Q= 
M, and that ( cos. — ^ — ) must be extremely near unity ; 

and, therefore, as a factor, may disappear ; we then have, 

„ . M (a4-h)r («4-5> 

2 81^-^- = ^ — i -^-, or, a?=: v ; ' ; a - 
2 x 2 sin. \M 

A more ancient method is to compute the value of the little 

triangle B C G, and then of the whole triangle AMG, and 

then of a part, A M C or M G C. 



68 ASTRONOMY. 

Ohap^ti. (j 64.) Many times, when the moon comes round to its peri- 
gee, we find its parallax less than 61' 29", and, at the oppo- 
site apogee, more than 53' 50". It is only when the sun is 
in, or near a line with the lunar perigee and apogee, that 
these greatest extremes are observed to happen ; and when 
the sun is near a right angle to the perigee and apogee, then 
the moon moves round the earth in an orbit nearer a circle ; 
and thus, by observing with care the variation of the moon's 
parallax, we find that its orbit is a revolving ellipse, of variable 
eccentricity. 

(G5.) Because the moon's distance from the earth is va- 
riable, therefore there must be a mean distance: we shall 
show, hereafter, that her motion is variable; therefore there 
is a mean motion ; and, as the eccentricity is variable, there 
is a mean eccentricity. 
Mean pa- The extreme parallaxes, at mean eccentricity, are 60' 20", 
parallax ' at anc ^ — ^" ' an( ^ * ae corresponding distances from the earth 
mean dis- are 56.93 and 63.64; the radius of the earth being unity. 
tance> The mean parallax, or mean between 60' 20" and 54' 05", is 

57' 12". 5; but the parallax, at mean distance, is 57' 03"*. 



* It may seem paradoxical that the mean parallax, and the 
parallax at mean distance are different quantities ; but the 
following investigation will set the matter at rest. Let d and 
D be extreme distances, and M the mean distance. 

Then, - - - - d+D=2M; . . '. (1) 

Also, let p and P be the parallaxes corresponding to the dis- 
tances d and D ; and put x to represent the parallax at mean 
distance. Then, by Art. 60 ( if we call the radius of the 
tables unity), we have 

11 1 

d=- , D = - =:, and M-- 



sin. p sin. P' sin. x 

Substituting these values of d, D, and M, in equation (1) we 

112 

have, - - -f~ 



sin.^ sin. P sin. x 

Or, - - - sin. P + sin. p = f (z) 

1 ■■■•■■■ •. sin. x 



VARIATION OF PARALLAX. 



69 



The mean between extreme distances is 



55.92+63-84 
2 



or 



59.88 



Chap. II. 



rallax. 



but the true mean distance is 60.26, corresponding to the Mean di3_ 
parallax 57' 3". The mean, between extremes, is a variable moQn 
quantity ; but the true mean distance is ever the same ; a 
little more than 60i times the semidiameter of the earth. 

(66.) The variations in the moon's real distance must cor- 
respond to apparent variations in the moon's diameter ; and if 
the moon, or any other body, should have no variation in 
apparent diameter, we should then conclude that the body 
was always at the same distance from us. 

The change, in apparent diameter, of any heavenly body, is 
numerically proportioned to its real change in distance ; as 
appears from the demonstration in the note below.* 

But by a well known, and general theorem in trigonometry, Mean pa- 

we have, sin. P+sin.^=2 sin.f — ~-~^- ) cos. ( 9 ) (3) 

By equating (3) and (2), and observing that the cosines 
of very small arcs may be practically taken as unity, or ra- 
dius, therefore, 

/P-\-p\ sin. P sin. p 

sin. I — +^- ) = : -- ; 

\ 2 / sin. x 

>l . sin. P sin. p 

Or, sin. x = -r— — — — -. 

sm±(P-\-p) 

On applying this equation, we find #=57' 3". 

* Let A be the Fi S- 15 - 

point of vision, and 
d the diameter of 
any body at diffe- 
rent distances,^ B, 

AC 1_ Be 

Now, by trigonometry, we have the following proportions : 
AC : d : : R : tan. CAD 
AB : d :: B : tan. BAE. 




70 ASTRONOMY. 

Chaf ' u - Now if the moon has a real change in distance, as observa- 
tions show, such change must he accompanied with apparent 
changes in the moon's diameter ; and, by directing observa- 
tions to this particular, we find a perfect correspondence ; 
showing the harmony of truth, and the beauties of real 
science. 
Connec- We have several times mentioned that the moon's horizon- 
tion etween ^j p ara i] ax } g fa e semidiameter of the earth, as seen from the 

semidiame- *- 7 

ter and hori- moon ; and now we further say, that what we call the moon's 
zontai parai- gemidiameter, an observer at the moon would call the earth's 

;ax„ 

horizontal parallax; and the variation of these two angles de- 
pends on the same circumstance — the variation of the distance 
between the earth and moon; and, depending on one and the 
same cause, they must vary in just the same proportion. 

When the moon's horizontal parallax is greatest, the moon's 
semidiameter is greatest ; and, when least, the semidiameter 
is the least ; and if we divide the tangent of the semidiameter 
by the tangent of its horizontal parallax, we shall always find 
the same quotient (the decimal 0.27293); and that quotient 
is the ratio between the real diameter of the earth and the 
diameter of the moon.* Having this ratio, and the diameter 
of the earth, 7912 miles, we can compute the diameter of the 

moon thus : 

7912x0.27293=2169.4 miles. ^ 

From the first proportion, - - - AC tan. CAD=dB; 

From the second, AB tan. BAE=dR ; 

By equality, - - - - A (7tan. CAD=AB tan. BAE. 
This last equation, put into an equivalent proportion, gives : 
AC : AB : tan. BAE :: tan. CAB. 

But tangents of very small arcs ( such as those under which 
the heavenly bodies appear) are to each other as the arcs 
themselves. Therefore, 

AC \ AB w Singh BAE : angle CAD; 
That is; the angular measures of the same body are inversely 
proportional to the corresponding distances. 

* This requires demonstration. Let E be the real semi- 



APPEARANCE FROM THE MOON 



71 



As spheres are to each other in proportion to the cubes of Chap, ii, 
their diameters, therefore the bulk ( not mass ) of the earth, 
is to that of the moon, as 1 to T \, nearly. 

As the moon's distance is 60 i times the radius of the earth, Augmen- 
it follows that it is about JUfc nearer to us, when at the tation of the 

° • . . moons semi- 

zenith, than when in the horizon. Making allowance for this diameter : its 

(in proportion to the cosine of the altitude), is called the cause - 

augmentation of the semidiameter. 

(68.) It may be remarked, by every one, that we always The earth 

see the same face of the moon ; which shows that she must a moon t0 

• • • -i i • -i ^e moon - 

roll on an axis in the same time as her mean revolution about 

the earth ; for, if she kept her surface toward the same part 

of the heavens, it could not be constantly presented to the 

earth, because, to her view, the earth revolves round the 

moon, the same as to us the moon revolves round the earth ; 

and the earth presents phases to the moon, as the moon does 

to us, except opposite in time, because the two bodies are 

opposite in position. When we have new moon, the lunarians 

have full earth ; and when we have first quarter, they have 

last quarter, etc. The moon appears, to us, about half a 

degree in diameter ; the earth appears, to them, a moon, about 



Fig. 16. 



diameter of 
the ea r t h 
(Fig. 16), wi 
that of the 
moon, D the 
distance be- 
tween the 

two bodies ; and let the radius of the tables be unity. Put 
P to represent the moon's horizontal parallax, and s its appa- 
rent semidiameter. Then, by trigonometry, 

i D : E : : 1 : tan. P : and D : m : : 1 : tan. s. 




From the first, D= 



E 



tan.i>' 



from the 2d, D= 



m 



tan. s ' 



Therefore, 
6 



E 



m 



m 



tan. P tan. $ 



or 



tan. s 



tan. P E' 



Q. E.D. 



revolves on 
an axis 



72 ASTRONOMY. 

Chap. ii. two degrees in diameter, invariably fixed in their shy, and the 
stars passing slowly behind it. 

The moon " But," says Sir John Herschel, "the moon's rotation on 
her axis is uniform ; and since her motion in her orbit is not 
so, we are enabled to look a few degrees round the equatorial 
parts of her visible border, on the eastern or western side, 
according to circumstances; or, in other words, the line join- 
ing the centers of the earth and moon fluctuates a little in its 
position, from its mean or average intersection with her sur- 
face, to the east, or westward. And, moreover, since the 
axis about which she revolves is not exactly perpendicular to 
her orbit, her poles come alternately into view for a small 
space at the edges of her disc. These phenomena are known 
by the name of librations . In consequence of these two dis- 
tinct kinds of libration, the same identical point of the moon's 
surface is not always the center of her disc ; and we therefore 
get sight of a zone of a few degrees in breadth on all sides 
of the border, beyond an exact hemisphere.' 



CHAPTER III. 

THE EARTH'S ORBIT ECCENTRIC. THE APPARENT ANGULAR 

MOTION OP THE SUN NOT UNIFORM. LAWS BETWEEN DIS- 
TANCE, REAL, AND ANGULAR MOTION. ECCENTRICITY OF 

THE ORBIT. 

Chaf - 1I f ( 69. ) The sun's parallax is too small to be detected by 
The sun any common means of observation ; hence it remained un- 
known, for a long series of years, although many ingenious 
methods were proposed to discover it. The only decision 
that ancient astronomers could make concerning it was, that 
it must be less than 20" or 15" of arc ; for, were it as much 
as that quantity, it could not escape observation. 

Now let us suppose that the sun's horizontal parallax is less 
than 20" ; that is, the apparent semidiameter of the earth, as 
seen from the sun, must be less than 20"; but the semidia- 



larger than 
the earth 



APPARENT DIAMETERS. 73 

meter of the sun is 15' 56", or 956" ; therefore the sun must Chap. hi. 

be vastly larger than the earth — by at least 48 times its 

diameter ; and the bulk of the earth must be, to that of the 

sun, in as high a ratio as 1 to the cube of 48. But as we do 

not suffer ourselves to know the true horizontal parallax of 

the sun, all the decision we can make on this subject is, that 

the sun is vastly larger than the earth. 

( 70. ) Previous observations, as we explained in the first Does the 
section of this work, clearly show, or give the appearance of * un g0 r ° un r 
the sun going round the earth once in a year ; but the appear- the earth 
ance would be the same, whether the earth revolves round the round the 

sun ? 

sun, or the sun round the earth, or both bodies revolve round 
a point between them. We are now to consider which is the 
most probable : that a large body should circulate round a much 
smaller one; or, the smaller one round a large one. The last 
suggestion corresponds with our knowledge and experience in 
mechanical philosophy ; the first is opposed to it. 

(71.) We have seen, in the last chapter, that the semidia- 
meter and horizontal parallax of a body have a constant rela- 
tion to each other; and, while we cannot discover the one, 
we will examine all the variations of the other ( if it have va- 
riations ), and thereby determine whether the earth and sun 
always remain at the same distance from each other. 

Here it is very important that the reader should clearly Methods 
understand, how the apparent diameter of a heavenly body of measunr >g 

. . . apparent di a- 

can be determined to great precision. meters. 

As an example, we shall take the diameter of the sun ; but 
the same principles are to be followed, and the same deduc- 
tions are to be made, whatever body, moon, or planet, may be 
under observation. 

An instrument to measure the apparent diameter of a planet The micro 
is called a micrometer. It is an eyepiece to a telescope, with meter - 
opening and closing parallel wires ; the amount of the opening 
is measured by a mathematical contrivance. For the measure 
of all small objects, the micrometer is exclusively used; and 
since it is impossible that any one observation can be relied 
upon as accurate (on account of the angular space eclipsed 
by the wires), a great number of observations are taken, and 

G 



74 ASTRONOMY. 

Chap, hi. t"he mean result is regarded as a single observation. Gene- 
rally speaking, the following method is more to be relied upon, 
when large angles are measured, and to it we commend special 
attention. 
The me- The method depends on the time employed by the body in pass- 

in ° y ! m ^ *W fl ie perpendicular wires of the transit instrument. 

the meridian. All bodies, (by the revolution of the earth) come to the 
meridian at right angles, and 15 degrees pass by the meridian 
in one hour of sidereal time; and, in four minutes, one de- 
gree will pass; and, in two minutes of time, 30 minutes of arc 
will pass the meridian wire. 

Now if the sun is on the equator, and stationary there, and 
employs two minutes of sidereal time in passing the meridian, 
then it is evident that its apparent diameter is just 30' of arc; 
if the time is more than two minutes, the diameter is more ; 
if less, less. 

But we have just made a supposition that is not true ; we 
have supposed the sun stationary, in respect to the stars ; but 
it is not so ; it apparently moves eastward ; therefore it will 
not get past the meridian wire as soon as it would if station- 
ary. Hence we must have a correction, for the sun's motion, 
applied to the time of its passing the meridian. 
Corrections We have also supposed the sun on the equator, and for a 

to be made. . . . . i ... -it ..,. 

moment continue the supposition, and also conceive its dia- 
meter to be just 30' of arc. Now suppose it brought up to 
the 20th degree of declination, on that parallel, it will extend 
over more than 30' of arc, because meridians converge toward 
the pole ; therefore the farther the sun, or any other body is from 
the equator, the longer it will be in passing the meridian on that 
account ; the increase of time depending on the cosine of the 
declination. (See 59.) 

Hence two corrections must be made to the actual time 
that the sun occupies in crossing the meridian wire, before we 
can proportion it into an arc ; one for the progressive motion 
of the sun in right ascension ; and one for the existing decli- 
nation. We give an example. 
Method of ® n *^e ^ rst ^ a y °? J une > 1846, the sidereal time ( time 
deciding the measured by the sidereal clock ) of the sun passing the me- 



APPARENT DIAMETERS. 75 

ridian wire, was observed to be 2 m. 16.64 s.; the declination Chap. hi. 



was 22° 2' 45", and the hourly increase of right ascension was exact appa . 

10.235 s. What was the sun's semidiameter? rent diame- 

ter of the 

3600 s. : 10.235 s. :: 136.64 : 0.39 s. bob, moon, 

or planets. 

Observed dura, of tran., in sees., 136.64 
Reduction for solar motion, .39 

136.25 . . log. 2.134337 
Dec. 22° 2' 45"; cosine, - - - 9.967021 

Duration, if stationary on equa., 126.3 s. . .log. 2.101358 

Minutes or seconds of time can be changed into minutes or 
seconds of arc, by multiplying by 15 ; therefore the diameter 
of the sun, at this time, subtended an arc of 1894". 5, and its 
semidiameter 947". 2, or 15' 47". 2 ; which is the result given 
in the Nautical Almanac, from which any number of examples of 
this kind can be taken. We give one more example, for the 
benefit of those who may not have a Nautical Almanac. 

On the 30th day of December (not material what year), 
the sidereal time of the sun's diameter passing the meridian 
was observed to be 2 m. 22.2 s., or 142.2 s. The sun's 
hourly motion in right ascension, at that time, was 11.06 s., 
and the declination was 22° 11'. What was the sun's semi- 
diameter?* Ans. 16' 17".3. 

These observations may be made every clear day through- Extreme 

values of the 

out the year ; and they have been made at many places, and SDn , s appa . 

for many years ; and the combined results show that the rent semidia- 
meter. 

* The following is the formula for these reductions : 
15(f— c)cos.D 
R = S - 

Here t is the observed interval in seconds, c is the correction for the in- 
crease in right ascension, D is the declination, R the radius of the tables, 
and s is the result in seconds of arc. c is always very small ; for one 
hour, or 3600 s., the variation is never less than 8.976 s., nor more than 
11.11 s. The former happens about the middle of September ; the lat- 
ter about the 20th of December. For the meridian passage of the moon, 
the correction c is considerable ; because the moon's increase of right 
ascension is comparatively very rapid. For the planets, c may be dis- 
regarded. 



76 ASTRONOMY. 

Ch ap, in . a pp aren t diameter of the sun is the same, on the same day of 
the year, from whatever station observed. 

The least semidiameter is 15' 45". 1 ; which corresponds, in 
time, to the first or second day of July ; and the greatest is 16' 
17". 3, which takes place on the 1st or 2d of January. 

Now as we cannot suppose that there is any real change in 

the diameter of the sun, we must impute this apparent change 

to real change in the distance of the body, as explained in 

Art. 66. 

Variation Therefore the distance to the sun, on the 30th of Decem- 

1 ie _ 1S " ber, must be to its distance, on the first day of July, as the 

tance from */••_«/» 

the earth to number 15' 45". 1 is to the number 16' 17".3, or as the num- 
the sun. ber 945.1 to 977.3; and all other days in the year, the pro- 
portional distance must be represented by intermediate num- 
bers. 

From this, we perceive, that the sun must go round the 
earth, or the earth round the sun, in very nearly a circle ; for 
were a representation of the curve drawn, corresponding to 
the apparent semidiameter, in different parts of the orbit, and 
placed before us, the eye could scarcely detect its departure 
from a circle. 

( 72.) It should be observed that the time elapsed between 
the greatest and least apparent diameter of the sun, or the 
reverse, is just half a year ; and the change in the sun's lon- 
gitude is 180°. 
Eccentri- If we would consider the mean distance between the earth 
earth's orbit 6 an< ^ sun as un ^V ( as * s customary with astronomers), and then 
how known, put x to represent the least distance, and y the greatest dis- 
tance, we shall have 

x-\-y=2. 

And, - - x : y :: 9451 : 9773. 

A solution gives #=0.98326, nearly, and y=l. 01674, nearly: 
showing that the least, mean, and greatest distance to the sun, 
must be very nearly as the numbers .98326, 1., and 1.01674. 
The fractional part, .01674, or the difference between the 
extremes and mean ( when the mean is unity ), is called the 
eccentricity of the orbit. 



SUN'S MOTION IN LONGITUDE. 77 

The eccentricity, as just mentioned, must not be regarded as chap. ni. 
accurate. It is only a first approximation, deduced from the 
first and most simple view of the subject ; but we shall, here- 
after, give other expositions that will lead to far more accu- 
rate results. 

In theory, the apparent diameters are sufficient to determine Eccentrici- 
the eccentricity, could we really observe them to rigorous ^ re ^° m d ^- 
exactness : but all luminous bodies are more or less affected meters only 
by irradiation, which dilates a little their apparent diameters ; a PP roximate - 
and the exact quantity of this dilatation is not yet well 
ascertained. 

( 73. ) The sun's right ascension and declination can be 
observed from any observatory, any clear day; and from 
thence we can trace its path along the celestial concave sphere 
above us, and determine its change from day to day ; and we 
find it runs along a great circle called the ecliptic, which 
crosses the equator at opposite points in the heavens ; and 
the ecliptic inclines to the equator with an angle of about 
23° 27' 40". 

The plane of the ecliptic passes through the center of the 
earth, showing it to be a great circle, or, what is the same 
thing, showing that the apparent motion of the sun has its 
center in the line which joins the earth and sun. 

The apparent motion of the sun along the ecliptic is called Variations 

longitude ; and this is its most regular motion. m the dls ' 

-vVn ,'-.-. . tance of the 

When we compare the sun s motion, m longitude, with its sun com . 

semidiameter, we find a correspondence — at least, an apparent P ared with 

. its variations 

connection. in longitude . 

When the semidiameter is greatest, the motion in longitude 
is greatest ; and, when the semidiameter is least, the motion 
in longitude is least ; but the two variations have not the same 
ratio. 

When the sun is nearest to the earth, on or about the 30th 
of December, it changes its longitude, in a mean solar day, 
1° 1' 9".95. When farthest from the earth, on the 1st of 
July, its change of longitude, in 24 hours, is only 57' 11".48. 
A uniform motion, for the whole year, is found to be 59' 8".33. 

The ancient philosophers contended that the sun moved 

G* 



78 ASTRONOMY. 

Chap, in . about the earth in a circular orbit, and its real velocity uni- 
form ; but the earth not being in the center of the circle, the 
same portions of the circle would appear under different angles ; 
and hence the variation in its apparent angular motion. 
The result ]\j ow jf ^his is a true view of the subject, the variation in 

shows that . " . . 

the angular an gular motion must be m exact proportion to the variation in 
motion is in distance, as explained in the note to Art. 66; that is, 945". 1 

pro'porUor 6 Sh0uld be t0 977 "- 8 > aS 57 ' n "- 48 t0 61 ' 9 "- 95 > if the SU P" 

to the square position of the first observers were true. But these numbers 
1S " have not the same ratio ; therefore this supposition is not 



tance. 



satisfactory ; and it was probably abandoned for the want of 
this mathematical support. The ratio between 945". 1, and 

Q77Q 

977".3is .JJL = 1.0341, nearly; 

9451 J 

266Q" Q'S 
between 57' 11".48, and61'9".95, ^_4r = 1.0694, nearly. 

3431".48 J 

If we square (1.0341) the first ratio, we shall have 1.06936, 
a number so near in value to the second ratio, that we con- 
clude it ought to be the same, and would be the same, pro- 
vided we had perfect accuracy in the observations. 
Law be- Thus we compare the angular motion of the sun in diffe- 
tion and dis- ren * P ar ^ s °f its orbit ; and we always find, that the inverse 
ance. square of its distance is proportional to its angular motion; and 

this incontestible/actf is so exact and so regular, that we lay 
it down as a law ; and if solitary observations do not corre- 
spond with it, we must condemn the observations, and not 
the law. 

(74.) To investigate this subject thoroughly, we cannot 
avoid making use of a little geometry. 

Let Pig. 17 represent the solar orbit,* the sun apparently 
revolving about the observer at 0. The distance from to 



* We say solar orbit, when it is really the earth's orbit ; so we speak 
of the sun's motion, when it is really the motion of the earth ; and it 
is customary, with astronomers, to speak of apparent motions as real • 
and none object to this manner of speaking, who have a clear or en- 
larged view of the science — for to depart from it would lead to oft- 
repeated and troublesome technicalities, if not to confusion of ideas. 
Clearness does not always correspond with exactness of expression. 



VARIATIONS IN SOLAR MOTION. 



79 



amy point in the or- 
bit is called the ra- 
dius vector; and it is 
a varying quantity, 
conceived to sweep 
round the point 0. 
Let D be the va- 
lue of the radius vec- 
tor at any point, and 
rD its value at some 
other point, as repre- 
sented in the figure. 



Fig. 17. 



Chap. in. 




Let y represent the real motion of the Variations 



sun. for a very short interval of time, at the extremity of the m , and 

" ° angular mo- 

radius vector D ; and x represent the real motion, at the tion. 
extremity of the radius vector r D, in the same time. 

From 0, as a center, at the distance of unity, describe a 
circle. Put A to represent the angle under which x appears 
from 0; then, by observation, r 2 A is the angle under which y 
appears from the same point. 

Now, considering the sectors as triangles, we have the fol- 
lowing proportions : 

: A : : rD : x; 
: r 2 A : : D : y. 

- x=rAD, 

y=r 2 AD. 

Multiply the first of these equations by r, and we perceive 
that ------ y=rx. 

This last equation shows that the real velocity of the earth The real 



1 
1 

From the first, • 

From the second, 



or 



velocity of 
the earth in 
its orbit va- 
ries as the 
sun's appa- 



in its orbit varies in the inverse ratio as the radius vector 
it varies directly as the apparent diameter of the sun. 

(75.) If we multiply r D hjx, the product will express the 
double of an area passed over by the radius vector in a certain rent diame 
interval of time ; and if we multiply D by y, we shall have ter> 
the double of another area passed over by the radius vector in 
the same time. But the first product is rDx, and the second 
is the same, as we shall see by taking the value of y (r x) ; that 
is r D x=r B x \ hence we announce this general law: 



motion in an 

ellipse. 



80 ASTRONOMY. 

C hap, hi . That the solar radius vector describes equal areas in equal 

The radius times. 

.ec or e- ^Yh en expressed in more general terms, this is one of the 

serines equal I o ' 

areas in e- three laws of Kepler, which will he fully brought into notice 
qua time,. j Q a su ] 3ge q lien t p ar £ f this work. 

If we draw lines from any point in a plane, reciprocally 
proportional to the sun's apparent diameter, and at angles 
differing as the change of the sun's longitude, and then con- 
nect the extremities of such lines made all round the point, 
the connecting lines will form a curve, corresponding with an 
ellipse (see Fig. 18), which represents the apparent solar orbit ; 
and, from a review of the whole subject, we give the follow- 
ing summary : 
Laws of 1. The eccentricity of the solar ellipse, as determined from the 
apparent diameter of the sun, is .01674.* 

2. The sun's angular velocity varies inversely as the square 
of its distance from the earth. 

8. The real velocity is inversely as the distance. 

4. The areas described by the radius vector are proportional 
to the times of description. 

(76.) We have several times mentioned, that, as far as 
appearances are concerned, it is immaterial whether we con- 
sider the sun moving round the earth, or the earth round the 
sun; for, if the earth is in one position of the heavens, the 

* By making use of the 2d principle, above cited, we can 
compute the eccentricity of the orbit to greater precision than 
by the apparent diameters, because the same error of obser- 
vation on longitude, would not be as proportionally great as 
on apparent diameter. 

Let E be the eccentricity of the orbit; then (1 — E) is 
the least distance to the sun, and (l-\-E) the greatest dis- 
tance. Then, by observation, we have 



(1—E) 2 : (1+Ef- 
Or, (1— Ey : (1+Ey 
Or, 1—E : \-\-E 



57' 11".48 
343148 



^343148 



61' 9".95; 
366995 s 



^366995. 



Whence ^=.016788-)-. We shall give a still more accu- 
rate method of computing this important element. 



SUN'S ELLIPTICAL MOTION 



81 




Chap. Ill, 



sun appears exactly in 
the opposite position, 
and e v e r y ra o t i o n 
made by the earth 
must correspond to an 
apparent motion made 
by the sun. 

But, for the purpose 
of getting nearer to 
fact, we will now sup- 
pose the earth revolves round the sun in an elliptical orbit, 
as represented by Fig. 18. 

We have very much exaggerated the eccentricity of the 
orbit, for the purpose of bringing principles clearer to view. 

The greatest and least distances, from the sun to the earth, 
make a straight line through the sun, and cut the orbit into 
two equal parts. When the earth is at B, the greatest dis- 
tance from the sun, it is said to be in apogee, and when at A, 
the least distance, it is in perigee ; and the line joining the 
apogee and perigee is the major, or greater diameter of the 
orbit ; and it is the only diameter passing through the sun, thai 
cuts the orbit into tivo equal parts. 

Now, as equal areas are described in equal times, it follows 
that the earth must be just half a year in passing from apogee tl0ns t0 ®' 
to perigee, and from perigee to apogee ; provided that these positions of 
points are stationary in the heavens, and they are so, very the solar a 

pogee 

nearly.* perigee 

If we suppose the earth moves along the orbit from D to 
A, and we observe the sun from D, and continue observa- 
tions upon it until the earth comes to C, then the longitude 
of the sun has changed 180°; and if the time is less than 



Observa- 



and 



* The longer axis of the orbit, or apogee point, changes position by 
u very slow motion of about 12" per annum, to the eastward : but this 
motion must be disregarded, for the present, as well as many other mi- 
nute deviations, to be brought into view when we are better prepared 
to understand them. 

Those minute variations, for short periods of time, do not sensibly 
affect general results. 

6 



82 ASTRONOMY. 

Cha>. in. lialf a year, we are sure the perigee is in this part of the 
orbit. If we continue observations round and round, and 
find where 180 degrees of longitude correspond with half a 
year, there will be the position of the longer axis ; which is 
sometimes called the line of the apsides. 

Difficulties, vy e cannot determine the exact point of the apogee or 
perigee, by direct observations on the sun's apparent diame- 
ter; for about these points the variations are extremely slow 
and imperceptible. 

If we take observations in respect to the sun's longitude, 
when the earth is at b, and watch for the opposite longitude, 
when the earth is about a, and find that the area b Da was 
described in little less than half a year, and the area a C b, in 
a little more than half a year, then we know that b is very 
near the apogee, and a very near the perigee. 

If we take another point, b', and its opposite, a, and find 
converse results, then we know that the apogee is between 
the points b' and b, and we can proportion to it, to great exact- 
ness. 
Longitude ( 77. ) The longitude of the apogee, for the year 1801, was 

Ld erkee 6 ^° ^' ^ "' an< ^ °^ course > the perigee was in longitude 279° 
31' 9". These points move forward, in respect to the stars, 
about 12" annually, and, in respect to the equinox, about 62" ; 
more exactly 61". 905, and, of course, this is their annual 
increase of longitude. 

In the year 1250, the perigee of the sun coincided with the 
winter solstice, and the apogee with the summer solstice ; and 
at that time the sun was 178 days, and about 17-|- hours, on 
the south side of the equator, and 186 days, and about 12^ 
hours, on the north side ; being longer in the northern hemi- 
sphere than in the southern, by seven days and 19 hours: at 
present, the excess is seven days and near 17 hours. 
The year / "jg ^ As the gun } s a i onger time in the northern than in 

unequally di- ., .. i • i i • • i -i n 

vided. ^ ne southern hemisphere, the first impression might be, that 

more solar heat is received in one hemisphere than in the 
other ; but the amount is the same ; for whatever is gained 
in time, is lost in distance ; and what is lost in time, is gained 
by a decrease of distance. The amount of heat depends on 



SUN'S ELLIPTICAL MOTION. 83 

the intensity multiplied by the time it is applied ; and the Chap. hi. 
product of the time and distance to the sun, is the same in 
either hemisphere; but the amount of heat received, for a 
single day, is different in the two hemispheres. 

(79.) Conceive a line drawn through the sun, at right 
angles to the greater diameter of the orbit D S C ( see Fig. 
18), the point C is 8° 21' from the first point of Aries; and 
if we observe the time occupied by the sun in describing 180 
degrees of longitude, from this point (or from any point very 
near this point), that time, taken from the whole year, will 
give the time of describing the other 180 degrees. 

Without being very minute, we venture to state, that the a method 
time of describing the arc DA C, is 178 days m hours; and ° f obtainin § 

® # - . . ' ■ 'the eccentri- 

the time of describing the arc CBD is 186 days 12i hours, city of an or- 
But, as areas are in proportion to the times of their descrip- bit - 
tion; therefore, 

d. h. d. h. 

area CSDA : area CBDS : : 178 17* : 186 12i. 

By taking half of the greater axis of the ellipse equal 
unity, and the eccentricity an unknown quantity, e, the 
mathematician can soon obtain analytical expressions for 
the two areas in question, and then, from the proportion, 
he can find the value of the eccentricity e: but there is a 
better method — we only give an outside view of this, for the 
ligM it throws on the general principle. 

( 80.) Now let us conceive the orbit of the earth inclosed 
by a circle whose diameter is the greatest diameter of the 
ellipse, as represented by Fig. 19. 

For the sake of simplicity we will suppose the observer at Prepara- 
rest at the point o ( one focus of the ellipse ), and the sun tl0n for find ' 

. . ing the true 

really to move round on the ellipse, describing equal areas variation in 
in equal times round the point o. an elli P se - 

Conceive, also, an imaginary sun to pass round the circle, 
describing equal angles, in equal times, round the center m. 
Now suppose the two suns to be together at the point B ; — 
they depart, one on the ellipse, the other on the circle ; and, 
on account of both describing equal areas, in equal times, 
round their respective centers of motion, they will be together 



84 



ASTRONOMY. 




Fi g- 19 - at the point A, and 

again at the point B, 
and so continue in 
each subsequent re- 
volution. 

The imaginary sun 
B on the circle every- 
where describes equal 
angles in equal times ; 
and the true sun, on 
the ellipse, describes 
only equal areas in 
equal times ; but the angles will be unequal. Conceive the 
two suns to depart, at the same time, from the point B, 
and, after a certain interval of time, one is at s, the other at 
*'. Then we must have 

area oBs : area mBs' : : area ellipse : area circle. 
Mean and The angle Bms' is the angle the sun would make, or its 
increase in longitude from the apogee ; provided the angular 
motion of the sun was uniform. The angle Bos is its true 
increase of longitude ; the difference between these two angles 
is the angle m n o. 

The angle Bms' is always known hy the time ; and if to 
every degree of the angle B m s' we knew the corresponding 
angle m n o, the two would give us the angle Bos; for, 
Bins' — mno=mo n; or, Bos. 

The angle Bms' is called the mean anomaly, and the angle 

Bos is called the true anomaly. 

The equa- The angle Bms' is greater than the angle Bos, all the 

tion of the wa y f rom the apogee to the perigee; but from the perigee to 

the apogee the true sun, on the ellipse, is in advance of the 

imaginary sun on the circle. 

The angle mn o is called the equation of the center ; that is, 
it is the angle to be applied to the angle about the center m, 
to make it equal to the true anomaly. 

The angle mno depends on the eccentricity of the ellipse ; 
and its amount is put in a table corresponding to every 



tree 
maiy 



ECCENTRICITY OF ORBIT 



85 



degree of the mean anomaly ; subtractive, from the apogee to ceap. hi. 
the perigee, and additive from the perigee to the apogee.* 

(81.) Again; conceive the two suns to set out from the same The great- 
point. B • and as the angle B m s' increases uniformly, it -will est ec i uatl0n 

f ' a r> of the center 

increase and become greater and greater than the angle Bos, gi ves the ec- 
until the true sun attains its mean angular motion, and no cent ricity of 

mi • • IT ^ le oro 't. 

longer. Then the angle mno attains its greatest value, and, 
at that time the side mn=no, and the point n is perpendicular 
over om, and os is a mean proportional between o B and oA. 
That is, when the sun, or any planet, attains the greatest equa- 
tion of the center, the true sun is very near the extremity of 
the shorter axis of the ellipse : o, the greatest equation of the 
center, can be determined by observation ; and, from the 
greatest equation, we have the most accurate method of com- 
puting the eccentricity of the ellipse, as we may see by the 
note below.f 

t Let C (Fig. 20) be the 
place of the true sun, and G 
the place of the imaginary 
sun ; the line m F cuts off 
equal portions of the circle 
and the ellipse. Then we 
have, to make the sector 
m F G to the triangle o m C, 
as the circle is to the ellipse. Now let 

mB=a, mC=b, om=ea, cr=3.1416; 

Then, the area of the circle is ^a 2 ; the area of the ellipse is 

Trah ; that of the sector is ( GF)-, and of the triangle -f-. 




Hence, 



eab 

~9~ 



GF 



a 
2> 



rah 



Tra 2 



* By a mere mechanical contrivance, the modern astronomical tables 
are so arranged, that all corrections are rendered additive ; so that the 
mechanical operator cannot make a mistake, as to signs, and he may 
continue to work without stopping to think. These arrangements 
have their advantages, but they cover up and obscure principles. 

H 



86 ASTRONOMY. 

Chap. in. When once the eccentricity of any planetary ellipse be- 
comes known, the equation of the center, corresponding to all 
degrees of the mean anomaly, can be computed and put into 
a table for future use ; but this labor of constructing tables 
belongs exclusively to the mathematician. 



Method of 



Or, - - eab : (GF)a : : b : a; 



deducing the 

eccentricity Or, - - ea I GF : : 1 : 1 ; 
neatest e- Consequently, GF=ea, and FG=om ; which shows that the 
quation of angle o Cm is nearly equal to Fm G, unless it is a very eccen- 
tric ellipse. Now we must compute the number of degrees 
in the arc FG. The whole circumference is 2?ra. 

Therefore, 2jra : ea : : 360 : arc FG', 
Hence, - - - arc FG= =angle nmC. 

rr 

But the angle onm=nm C-\-n Cm=2nm C, nearly; 

Therefore, =2nm C=onm = greatest equation of 

center, nearly. 

But the greatest equation of the center, for the solar orbit, 
is, by observation, l c 55' 30" ; and as the sun has not quite 
its greatest equation of the center, when at the point C, it will 
be more accurate to put 

360 e „-_ __, nMI , 
=1° 55' 24". 

From this equation, it is true, we have only the approxi- 
mate value of e ; but it is a very approximate value, and suffi- 
ciently accurate. 

Keducing both members to seconds, and we have, 
3600-360 6=6924**, and e=0.0167842. 

The greatest equation of the center is at present diminish- 
ing at the rate of 17". 17 in one hundred years ; this corre- 
sponds to a diminution of eccentricity by 0.00004166; which 
is determined by a solution of the following equation : 

3G 00.^=17".17. 



CHANGE OF SEASONS. 



87 



CHAPTER IV. 

THE CAUSES OF THE CHANGE OF SEASONS. 

( 82. ) The annual revolution of the earth in its orbit, chap, iv. 
combined with the position of the earth's axis to the plane 
of its orbit, produces the change of the seasons. 

If the axis were perpendicular to the plane of its orbit, The cause 
there would be no change of seasons, and the sun would then ° tnec an s 8 

° of seasons. 

be all the while in the celestial equator. 

This will be understood by Fig. 21. Conceive the plane 
of the paper to be the plane of the earth's orbit, and conceive 
the several representations of the earth's axis, JVS, to be in- 
clined to the paper at an angle of 66° 32'. 

Fig. 21. 




In all representations of JVS, one half of it is supposed to 
be above the paper, the other half below it. 

NS is always parallel to itself; that is, it is always in the 
same position* — always at the same inclination to the plane 



* Except minute variations, which it would be improper to notice in 
this part of the work. 

7 



88 ASTRONOMY. 

Chap, iv . f its orbit — always directed to the same point in the hea- 
vens, in whatever part of the orbit it may be. 

The plane of the equator, represented by Eq, is inclined to 
the plane of the orbit by an angle of 23° 28'. 

importance ;g y inspecting the figure, the reader will gather a clearer 

of inspecting . .. 

the figure, view ot the subject than by whole pages of description; he 
will perceive the reason why the sun must shine over the 
north pole, in one part of its orbit, and fall as far short of 
that point when in the opposite part of its orbit ; and the 
number of degrees of this variation depends, of course, on the 
position of the axis to the plane of the orbit. 
Position of Now conceive the line N S to stand perpendicular to the 
plane of the paper, and continue so ; then Eq would He on 



cause 



change of the paper, and the sun would at all times be in the plane of 
seasons. ^e e q ua t r, and there would be no change of seasons. If 

N S were more inclined from the perpendicular than it now 

is, then we should have a greater change of seasons. 

By inspecting the figure, we perceive, also, that when it is 

summer in the northern hemisphere, it is winter in the 

southern ; and conversely, when it is winter in the northern, 

it is summer in the southern. 

When a line from the sun makes a right angle with the 

earth's axis, as it must do in two opposite points of its orbit, 

the sun will shine equally on both poles ; and it is then in the 

plane of the equator ; which gives equal day and night the 

world over. 

Equal days and nights, for all places, happen on the 20th 

of March, of each year, and on the 22d or 23d of September. 

At these times the sun crosses the celestial equator, and is 

said to be in the equinox. 
The equi- The longitude of the sun, at the vernal equinox, is 0° ; and 
noctiai and at foe autumnal equinox, its longitude is 180°. 
p 0ir , ts . The time of the greatest north declination is the 20th of 

June ; the sun's longitude is then 90°, and is said to be at 

the summer solstice. 

The time of the greatest south declination is the 22d of 

December; the sun's longitude, at that time, is 270°, and 

is said to be at the winter solstice. 



CHANGES OF SEASONS. 89 

By inspecting the figure, we perceive, that when the earth Chap. it. 
is at the summer solstice, the north pole, P, and a conside- Long sea- 
rable portion of the earth's surface around, is within the en- sons cf sun - 

' . li^ht and 

lightened half of the earth ; and as the earth revolves on its ^rkness at 

axis J\ T S, this portion constantly remains enlightened, giving and about 

a constant day — or a day of weeks and months duration, epo 

according as any particular point is nearer or more remote 

from the pole; the pole itself is enlightened full six months 

in the year, and the circle of more than 24 hours constant 

sunlight extends to 23° 28' from the pole (not estimating the 

effects of refraction). On the other hand, the opposite, or 

south pole, S, is in a long season of darkness, from which it 

can be relieved only by the earth changing position in its 

orbit. 

" Now, the temperature of anv part of the earth's surface Tem P era 

,- r . . ture of the 

depends mainly, if not entirely, on its exposure to the sun's earth, 
rays. Whenever the sun is above the horizon of any place, 
that place is receiving heat ; when below, parting with it, by 
the process called radiation; and the whole quantities re- 
ceived and parted with in the year must balance each other 
at every station, or the equilibrium of temperature would not 
be supported. Whenever, then, the sun remains more than 
12 hours above the horizon, of any place, and less beneath, 
the general temperature of that place will be above the ave- 
rage ; when the reverse, below. As the earth, then, moves 
from A to B, the days growing longer, and the nights shorter 
in the northern hemisphere, the temperature of every part of 
that hemisphere increases, and we pass from spring to sum- 
mer, while at the same time the reverse obtains in the southern 
hemisphere. As the earth passes from B to C, the days and 
nights again approach to equality — the excess of temperature 
in the northern hemisphere, above the mean state, grows less, 
as well as its defect, in the southern ; and at the autumnal 
equinox, C, the mean state is once more attained. From 
thence to D, and, finally, round again to A, all the same phe- 
nomena, it is obvious, must again occur, but reversed, it being 
now winter in the northern, and summer in the southern 
hemisphere." 



extreme tern 
perature. 



90 ASTRONOMY. 

Chap. iv. The inquiry is sometimes made why we do not have the 
warmest weather about the summer solstice, and the coldest 
weather about the time of the winter solstice. 
Times of This would be the case if the sun immediately ceased to 
give extra warmth, on arriving at the summer solstice ; but 
if it could radiate extra heat to warm the earth three weeks, 
before it came to the solstice, it would give the same extra 
heat three weeks after ; and the northern portion of the earth 
must continue to increase in temperature as long as the sun 
continues to radiate more than its medium degree of heat 
over the surface, at any particular place. Conversely, the 
whole region of country continues to grow cold as long as 
the sun radiates less than its mean annual degree of heat 
over that region. The medium degree of heat, for the whole 
year, and for all places, of course, takes place when the sun 
is on the equator; the average temperature, at the time of 
the two equinoxes. The medium degree of heat, for our 
northern summer, considering only two seasons in the year, 
takes place when the sun's declination is about 12 degrees 
north ; and the medium degree of heat, for winter, takes place 
when the sun's declination is about 12 degrees south; and 
if this be true, the heat of summer will begin to decrease 
about the 20th of August, and the cold of winter must essen- 
tially abate on or about the 16th of February, in all northern 
latitudes. 



CHAPTER V. 

EQUATION OF TIME 



( 83.) We now come to one of the most important subjects 
in astronomy — the equation of time. 

Without a good knowledge of this subject, there will be 

constant confusion in the minds of the pupils ; and such is 

the nature of the case, that it is difficult to understand even 

the facts, without investigating their causes. 

Sidereal Sidereal time has no equation ; it is uniform, and, of itself, 

time perfect, perfect and complete. 



EQUATION OF TIME. 91 

The time, by a perfect clock, is theoretically perfect and Chap. tv. 
complete, and is called mean time. 

The time, by the sun, is not uniform,; and, to make it Soiai im- 
agree with the perfect clock, requires a correction — a quan- notv ' n,!0;i1 • 
tity to make equality; and this quantity is called the equa- 
tion of time.* 

If the sun were stationary in the heavens, like a star, it 
would come to the meridian after exact and equal intervals 
of time; and, in that case, there would be no equation of 
time. 

If the sun's motion, in right ascension, were uniform, then 
It would also come to the meridian after equal intervals of 
time, and there would still be no equation of time. But 
( speaking in relation to appearances ) the sun is not station- 
ary in the heavens, nor does it move uniformly ; therefore it 
cannot come to the meridian at equal intervals of time, and, 
of course, the solar days must be slightly unequal. 

When the sun is on the meridian, it is then apparent noon, Mean and 
for that day ; it is the real solar noon, or, as near as may be, z w° 
half way between sunrise and sunset; but it may not be 
noon by the perfect clock, which runs hypothetically true and 
uniform throughout the whole year. 

A fixed star comes to the meridian at the expiration of 
every 23 h. 56 m. 04.09 s. of mean solar time ; and if the sun 
were stationary in the heavens, it would come to the meridian 
after every expiration of just that same interval, But the 
sun increases its right ascension every day, by its apparent 
eastward motion ; and this increases the time of its coming 
to the meridian ; and the mean interval between its successive 
transits over the meridian is just 24 hours ; but the actual 
intervals are variable — some less, and some more than 24 
hours. 

On and about the 1st of April, the time from one meridian 
of the sun to another, as measured by a perfect clock, is 23 h. 
59 m. 52.4 s. ; less than 24 hours by about 8 seconds. Here, 
then, the sun and clock must be constantly separating. On 



jrent 
noon 



* In astronomy, the terra equation is applied to all corrections to 
convert a mean to its true quantity. 



92 ASTRONOMY. 

Chap. v. and about the 20th of December, the time from one meridian 

of the sun to another is 24 h. m. 24.3 s., more than 24 

seconds over 24 hours ; and this, in a few days, increases to 

minutes — and thus we perceive the fact of equation of time. 

Equation To detect the law of this variation, and find its amount, 

u" 16 - ie we must separate the cause into its two natural divisions. 

result of two L 

causes. 1. The unequal apparent motion of the sun along the ecliptic. 

2. The variable inclination of this motion to the ecliptic. 

If the sun's apparent motion along the ecliptic were uni- 
form, still there would be an equation of time ; for that mo- 
tion, in some parts of the orbit, is oblique to the equator, and, 
in other parts, parallel with it ; and its eastward motion, in 
right ascension, would be greatest when moving parallel with 
the equator. 

From the first cause, separately considered, the sun and 
clock would agree two days in a year — the 1st of July and 
the 30th of December. 

From the second cause, separately considered, the sun and 
clock agree four days in a year — the days when the sun 
crosses the equator, and the days he reaches the solstitial 
points. 

When the results of these two causes are combined, the 
sun and clock will agree four days in the year ; but it is on 
neither of those days marked out by the separate causes ; and 
the intervals between the several periods, and the amount of 
the equation, appear to want regularity and symmetry. 
Days in The four days in the year on which the sun and clock 
the year in a2 . ree that is, show noon at the same instant, are April 15th, 

which the & 

sun and June 16th, September 1st, and December 24th. 

clock agree. rpj^ g rea t e st amount, arising from the first cause, is 7 m. 
42 s., and the greatest amount, from the second cause, is 9 m. 
53 s. ; but as these maximum results never happen exactly at 
the same time, therefore the equation of time can never 
amount to 17 m. 35 s. In fact, the greatest amount is 16 m. 
17 s., and takes place on the 3d of November ; and, for a long 
time to come, the maximum value will take place on the same 
day of each year ; but, in the course of ages, it will vary in 
its amount and in the time of the year in which the sun and 



EQUATION OF TIME 



93 



common 
cause. 



clock agree, in- consequence of the slow and gradual change Chap, v. 
in the position of the solar apogee. (See Art. 77.) 

( 84. ) The elliptical form of the earth's orbit gives rise to The equa- 
the unequal motion of the earth in its orbit, and thence to the tl0n of the 

. -i t • sun's center, 

apparent unequal motion of the sun in the ecliptic; and this a nd the first 
same unequal motion is what we have denominated the first P art of tho 
cause of the equation of time. Indeed, this part of the equa- ti ^ e ' have 
tion of time is nothing more than the equation of the center a 
(80), changed into time at the rate of four minutes to a degree. 

The greatest equation for the sun's longitude ( 81, note ), 
is by observation 1° 55' 30"; and this, proportioned into 
time, gives 7 m. 42 s., for the maximum effect in the equation 
of time arising from the sun's unequal motion. When the sun 
departs from its perigee, its motion is greater than the mean 
rate, and, of course, comes to the meridian later than it other- 
wise would. In such cases, the sun is said to be slow — and 
it is slow all the way from its perigee to its apogee ; and fast 
in the other half of its orbit. 

For a more particular explanation of the second cause, we 
must call attention to Fig. 22. 

Let T25^ (Fig. 
22 ) represent the 
ecliptic, and c p C ^= 
the equator. 

By the first cor- 
rection, the apparent 
motion along the 
ecliptic is rendered ^ 
uniform : and the sun 
is then supposed to 
pass over equal spaces 
in equal intervals of 
time along the arc 
°p S 25 . But equal 

spaces of arc, on the ecliptic, do not correspond with equal 
spaces on the equator. In short, the points on the ecliptic 
must be reduced to corresponding points on the equator. 
For instance, the number of degrees represented by qp S, on 




94 ASTRONOMY. 

chap. v. the ecliptic, is greater than to the same meridian along the 
equator. The difference between T S and t S', turned into 
time, is the equation of time arising from the obliquity of the 
ecliptic corresponding to the point S. 

At the points T, 25, and =£t, and also at the southern 
tropic, the ecliptic and the equator correspond to the same 
meridian; but all other equal distances, on the ecliptic and 
equator, are included by different meridians. 
How to To compute the equation of time arising from this cause, 
compute the we mus t solve the spherical triangle <pS S' ; <vSis the sun's 
of the equa- longitude, and the angle at °p is the obliquity of the ecliptic, 
tion of time, and at >S" is a right angle. Assume any longitude, as 32°, 
35°, or 40°, or any other number of degrees, and compute 
the base. The difference between this base and the sun's 
longitude, converted into time, is the quantity sought corre- 
sponding to the assumed longitude ; and by assuming every 
degree in the first quadrant, and putting the result in a table, 
we have the amount for every degree of the entire circle, for 
all the quadrants are symmetrical, and the same distance from 
either equinox will be the same amount, 
what is The perfect clock, or mean time, corresponds with the 
meant hy sun equator; and as uniform spaces along the equator, near the 

fast and slow . . .-n . -.. .-, , 7 

of clock point c p, will pass over more meridians than the same num- 
ber of equal spaces on the ecliptic ; therefore the sun, at S, 
will he fast of clock, or come to the meridian before it is noon 
by the clock — and this will be true all the way to the tropic, 
or to the 90th degree of longitude, where the sun and clock 
will agree. In the second quadrant, the sun will come to the 
meridian after the clock has marked noon. In the third qua- 
drant the sun will again be fast ; and, in the fourth quadrant, 
again slow of clock. 

It will be observed, by inspecting the figure, that what the 
sun loses in eastward motion, by oblique direction near the 
equator, is made up, when near the tropics, by the diminished 
distances between the meridians. 

For a more definite understanding of this matter, we give 
the following table. 



EQUATION OF TIME. 



95 



Table showing the separate results of the two causes for the equa- Chap, v. 
tion of time, corresponding to every fifth day of the second 
years after leap year ; but is nearly correct for any year. 







1st cause. 


2d cause. 












Sun slow 
of Clock, 

m. s. 


Sun slow 
ofClock. 




1st cause. 
Sun fast. 


2d cause. 
Sun slow. 






m. s. 




m. S. 


m. s. 


January 5 


41 


5 8 


July 1 





3 32 




10 


1 22 


6 35 


7 


40 


5 8 




15 


2 2 


7 48 


12 


1 19 


6 35 




20 


2 41 


8 45 


17 


1 57 


7 48 




25 


3 19 


9 26 


22 


2 35 


8 45 




29 


3 56 


9 49 


28 


3 12 


9 26 


Feb. 


3 


4 30 


9 53 


Aug. 2 


3 47 


9 49 




8 


5 2 


9 40 


7 


4 21 


9 53 




13 


5 32 


9 9 


12 


4 52 


9 40 




18 


5 39 


8 23 


17 


5 22 


9 9 




23 


6 24 


7 22 


22 


5 50 


8 23 




28 


6 45 


6 9 


28 


6 14 


7 22 


March 


5 


7 3 


4 46 


Sept. 2 


6 36 


6 9 




10 


7 18 


3 15 


7 


6 56 


4 46 




15 


7 29 


1 39 


12 


7 12 


3 15 




20 


7 37 


sun fast 


17 


7 24 


1 39 




25 


7 42 


1 39 


23 


7 34 


sun fast 




30 


7 42 


3 15 


28 


7 40 


1 39 


April 


4 


7 40 


4 46 


Oct. 3 


7 42 


3 15 




9 


7 34 


6 9 


8 


7 40 


4 46 




14 


7 24 


7 22 


13 


7 34 


6 9 




19 


7 12 


8 23 


18 


7 24 


7 22 




24 


6 56 


9 9 


23 


7 12 


8 23 




30 


6 36 


9 40 


23 


6 56 


9 9 


May 


5 


6 14 


9 53 


Nov. 2 


6 36 


9 40 




10 


5 50 


9 49 


7 


6 14 


9 53 




15 


5 22 


9 26 


12 


5 50 


9 49 




20 


4 52 


8 45 


17 


5 22 


9 26 




26 


4 21 


7 48 


22 


4 52 


8 45 




31 


3 47 


6 35 


27 


4 22 


7 48 


June 


5 


3 12 


5 8 


Dec. 2 


3 47 


6 35 




10' 


2 35 


3 32 


7 


3 12 


5 8 




16 


1 57 


1 48 


12 


2 35 


3 32 




21 


1 19 


sun slow 


17 


1 57 


1 48 




26 


40 


1 48 


21 

26 


1 19 
40 


sun slow. 
1 48 



By this table, the regular and symmetrical result of each „ 
cause is visible to the eye ; "but the actual value of the equa- preceding 
tion of time, for any particular day, is the combined results tabIe - 
of these two causes. Thus, to find the equation of time for 
the 5th day of March, we look at the table and find that 



96 ASTRONOMY. 

Chap. v. The first cause gives sun slow, - - - 7m. 3 s. 
The second, " sun slow, - - - 4 46 



Their combined result (or algebraic sum) is 11 49 slow. 

That is ; the sun being slow, it does not come to the meridian 
until 11 m. 49 s. after the noon shown by a perfect clock ; but 
whenever the sun is on the meridian, it is then noon, apparent 
time ; and, to convert this into mean time, or to set the clock, 
we must add 11 m. 49 s. 
Use of the j$y inspecting the table, we perceive, that on the 14th of 
time. April the two results nearly counteract each other ; and con- 

sequently the sun and clock nearly agree, and indicate noon 
at the same instant. On the 2d of November the two results 
unite in making the sun fast ; and the equation of time is 
then the sum of 6 36, and 9 40, or 16 m. 16 s. ; the maximum 
result. 

The sun at this time being fast, shows that it comes to the 
meridian 16m. 16s. before twelve o'clock, true mean time; 
or, when the sun is on the meridian, the clock ought to show 
11 h. 43 m. 44 s. ; and thus, generally, when the sun is fast, we 
must subtract the equation of time from apparent time, to obtain 
mean time ; and conversely, when the sun is slow. 

As no clock can be relied upon, to run to true mean time, 
or to any exact definite rate, therefore clocks must be fre- 
quently rectified by the sun. We can observe the apparent 
time, and then, by the application of the equation of time, we 
determine the true mean time. 
a table for (.85.) As the sun has a particular motion, corresponding 
equation of to every particular point on the ecliptic, and, at the same 
tns on^the ^ me ^ * ne particular point on the ecliptic has a definite rela- 
snn's longi tion to the equator, therefore any point, as S (Fig. 22), on 
tu e can e ^ Q ec ]jp^ Cj nag the two corrections for the equation of time ; 
consequently a table can be formed for the equation of time, 
depending on the longitude of the sun; and such a table 
would be perpetual, if the longer axis of the solar orbit did 
not change its position in relation to the equinoxes. But as 
that change is very slow, a table of that kind will serve for 



PLANETARY MOTIONS. 97 

many years, with a very trifling correction, and such a table Csap. v. 
is to Tbe found in many astronomical works. 

It is very important that the navigator, astronomer, and Utility of 
clock regulator, should thoroughly understand the equation of f e ti ^g atl0n 
time; and persons thus occupied pay great attention to it; 
but most people in common life are hardly aware of its ex- 
istence. 



CHAPTER VI. 

THE APPARENT MOTIONS OF THE PLANETS. 



(86.) "We have often reminded the reader of the great Chap, vi. 
regularity of the fixed stars, and of their uniform positions in 
relation to each other ; and by this very regularity and con- 
stancy of relative positions, we denominate them fixed; but 
there are certain other celestial bodies, that manifestly change 
their positions in space, and, among them, the sun and moon 
are most prominent. 

In previous chapters, we have examined some facts con- Recapitu- 
cerning the sun and moon, which we briefly recapitulate, as atl0n ' 
follows : 

1. That the sun's distance from the earth is very great; 
but at present we cannot determine how great, for the want 
of one element — its horizontal parallax. 

2. Its magnitude is much greater than that of the earth. 

3. The distance between the sun and earth is slightly va- 
riable ; but it is regular in its variations, both in distance and 
in apparent angular motion. 

4. The moon is comparatively very near the earth; its 
distance is variable, and its mean distance and amount of 
variations are known. It is smaller than the earth, although, 
to the mere vision, it appears as large as the sun. 

The apparent motions of both sun and moon are always in 
one direction; and the variations of their motions are never 
far above or below the mean. other ceies- 

But there are several other bodies that are not fixed stars ; *»»! bodies. 



98 ASTRONOMY. 

chap. vi. and although not as conspicuous as the sun and moon, have 
been known from time immemorial. 

They appear to belong to one family; but, before the true 

system of the world was discovered, it was impossible to give 

any rational theory concerning their motions, so irregular 

and erratic did they appear ; and this very irregularity of 

their apparent motions induced us to delay our investigations 

concerning them to the present chapter. 

The plan- j n general terms, these bodies are called planets — and 

' there are several of recent discovery — and some of very 

recent discovery; but as these are not conspicuous, nor well 

known, all our investigations of principles will refer to the 

larger planets, Venus, Mars, Jupiter, and Saturn. We now 

commence giving some observed facts ■", as extracted from the 

Cambridge astronomy 

The mom- ( 87.) " There are few who have not observed a beautiful 

mg and even- g ^ ar i n ^ e west, a little after sunset, and called, for this rea- 

in<j star. 

son, the evening star. This star is Venus. If we observe it 
for several days, we find that it does not remain constantly 
at the same distance from the sun. It departs to a certain 
distance, which is about 45°, or ith of the celestial hemi- 
sphere, after which it begins to return; and as we can ordi- 
narily discern it with the naked eye only when the sun is 
below the horizon, it is visible only for a certain time imme- 
diately after sunset. By and by it sets with the sun, and 
then we are entirely prevented from seeing it by the sun's 
light. But after a few days, we perceive, in the morning, 
near the eastern horizon, a bright star which was not visible 
before. It is seen at first only a few minutes before sunrise, 
and is hence called the morning star. It departs from the 
sun from day to day, and precedes its rising more and more ; 
but after departing to about 45°, it begins to return, ami 
rises later each day ; at length it rises with the sun, and we 
cease to distinguish it. In a few days the evening star again 
appears in the west, very near the sun ; from which it departs 
in the same manner as before ; again returns ; disappears for 
a short time ; and then the morning star presents itself. 
These alternations, observed without interruption for more 



PLANETARY MOTION. 99 

fchan 2000 years, evidently indicate that the evening and Chap. vi. 
morning star are one and the same body. They indicate, also, 
that this star has a proper motion, in virtue of which it oscil- 
lates about the sun, sometimes preceding and sometimes fol- 
lowing it. 

These are the phenomena exhibited to the naked eye ; but 
the admirable invention of the telescope enables us to carry 
our observations much farther." 

( 88. ) On observing Venus with a telescope, the irradiation The 
is, in a great measure, taken away, and we perceive that it 
has phases, like the moon. At evening, when approaching the 
sun, it presents a luminous crescent, the points of which are 
from the sun. The crescent diminishes as the planet draws 
nearer the sun ; but after it has passed the sun, and appears 
on the other side, the crescent is turned in the other direction ; 
the enlightened part always toward the sun, showing that it 
receives its light from that great luminarv. The crescent 

i ■ ..-.', i „ ;, The chases 

now gradually increases to a semicircle, and finally, to a full fVerms and 
circle, as the planet again approaches the sun ; hut, as the its a PP aren t 
crescent increases, the apparent diameter of the planet diminishes ; hE 



pauses 
of Venus. 



lave corn 



and at every alternate approach of the planet to the sun, the sponding 
phase of the planet is full, and the apparent diameter snUl ; ° 
and at the other approaches to the sun, the crescent diminishes 
down to zero, and the apparent diameter increases to its 
maximum. When very near the sun, however, the planet is 
lost in the sunlight ; but at some of these intervals, between 
disappearing in the evening, and reappearing in the morning, 
it appears to run over the sun's disc as a round, black spot ; 
giving a fine opportunity to measure its greatest apparent 
diameter.* When Venus appears full, its apparent diameter 
is not more than 10", and when a black spot on the sun, it 
is 59". 8, or very nearly V. Hence its greatest distance must 
be, to its least distance, as 59". 8 to 10, or nearly as 6 to 1. 



* Astronomers do not measure the apparent diameters of 
the planets by the process described for the sun and moon, 
because they pass the meridian too quickly. Most of them will 
pass the meridian in a small fraction of a second. They use 



ways near 

the sun. 



Greatest 
elongation. 



100 ASTRONOMY. 

Chap. vi. ( 89. ) When we come to form a theory concerning the 
real motion of this planet, we must pay particular attention 
to the fact, that it is always in the same part of the heavens 
Venus ai- as the sun — never departing more than 47° on each side of 
it — called its greatest elongation. In consequence of being 
always in the neighborhood of the sun, it can never come to 
the meridian near midnight. Indeed, it always comes to the 
meridian within three hours 20 minutes of the sun, and, of 
course, in daylight. But this does not prevent meridian ob- 
servations being taken upon it, through a good telescope ; * 

a micrometer, which is two spider lines, always parallel, near 
the focus of a telescope, and so attached, by the mechanism of 
screws, as to open and close at pleasure. 

To understand its grade of adjustment, bring the two lines 
together, so as to form one line. Then take any object, 
whose angular diameter is known at that time, as the diame- 
ter of the sun, and open the lines so as just to take in its 
disc, counting the turns, and parts of a turn given to the 
index screw to open to this object. From this we can com- 
pute the angle corresponding to one turn, or to any part of a 
turn, of the index screw. 

Now if we wish to measure the apparent diameter of any 
planet, bring the lines together, and then open them, just to 
inclose the planet ; and the number of turns, or the part of a 
turn, given to the screw, will determine the result. 

This may not be the exact mechanism of every micrometer, 
but this is the principle of their construction. 

* Perhaps we ought to have informed the reader before, "that the 
stars continue visible through telescopes, during the day, as well as the 
night ; and that, in proportion to the power of the instrument, not only 
the largest and brightest of them, but even those of inferior luster, such 
as scarcely strike the eye, at night, as at all conspicuous, are readily 
found and followed even at noonday, — unless in that part of the sky 
which is very near the sun, — by those who possess the means of point- 
ing a telescope accurately to the proper places. Indeed, from the bot- 
toms of deep narrow pits, such as a well, or the shaft of a mine, such 
bright stars as pass the zenith may even be discerned by the naked eye; 
and we have ourselves heard it stated by a celebrated optician, that the 



PLANETARY MOTION. 101 

and, as to this particular planet, it is sometimes so bright as cha?. Vi. 
to be seen by*the unassisted eye in the daytime. 

( 90.) Even without instruments and meridian observations, Morion of 
the attentive observer can determine that the motion of Venus, >enus m re - 

spect to tsie 

in relation to the stars, is very irregular — sometimes its stars . 
motion is rapid — sometimes slow — sometimes direct — some- 
times stationary, and sometimes retrograde ; * but the direct 
motion prevails, and, as an attendant to the sun, and in its 
own irregular manner, as just described, it appears to tra- 
verse round and round among the stars. 

(91. ) But Yenus is not the only planet that exhibits the Mercury 
appearances we have iust described. There is one other, and similar m a11 

1 L " > appearances 

only one — Mercury ; a very small planet, rarely visible to the t0 Venus. 
naked eye, and not known to the very ancient astronomers. 
Whatever description we have given of Venus applies to Mer- 
cury, except in degree. Its variations of apparent diameter 
are not so great, and it never departs so far from the sun ; 
and the interval of time, between its vibrations from one side 
to the other of the sun, is much less than that of Venus. 

(92.) These appearances clearly indicate that the sun must be A conclu- 
de center, or near the center, of these motions, and not the earth ; 
and that Mercury must revolve in an orbit within that of Venus. 

So clear and so unavoidable were these inferences, that even 
the ancients (who were the most determined advocates for 
the immobility of the earth, and for considering it as the 
principal object in creation — the center of all motion, etc.) 
were compelled to admit them; but with this admission, they 
contended, that the sun moved round the earth, carrying 
these planets as attendants. 

(93.) By taking observations on the other planets, the an- The 
cient astronomers found them variable in their apparent diam- rent diame- 

earliest circumstance which drew his attention to astronomy, was the 
regular appearance, at a certain hour, for several successive days, of a 
considerable star, through the shaft of a chimney." — HerscheVs Astro- 
nomy. 

* In astronomy, direct motion is eastward among the stars : station- 
ary is no apparent motion, in respect to the stars ; and retrograde is a 
westward motion. 



102 ASTRONOMY. 

Chap. vi. eters, and angular motions ; so much so, that it was impossible: 

ters 6f the to reconcile appearances with the idea of a stationary point of 

planets are observation ; unless the appearances were taken for realities, 

and that was against all true notions of philosophy. 

The planet Mars is most remarkable for its variations; and 
the great distinction between this planet and Venus, is, that 
it does not always accompany the sun ; but it sometimes, yea, 
at regular periods, is in the opposite part of the heavens from 
the sun — called Opposition — at which time it rises about 
sunset, and comes to the meridian about midnight. 
The earth The greatest apparent diameter of Mars takes place when 
no e cen- j pi ane + j s m opposition to the sun, and it is then 17".l, and 

ter ot its mo- i tx: 

ion. its least apparent diameter takes place when in the neighbor- 

hood of the sun, and it is then but about 4", showing that the 
sun, and not the earth, is the center of its motion. 
Systematic The general motion of all the planets, in respect to the 

irregularities g £ arSj j s direct ; that is, eastward ; but all the planets that 
attain opposition to the sun, while in opposition, and for some 
time before and after opposition, have a retrograde motion — 
and those planets which show the greatest change in appa- 
rent diameter, show also the greatest amount of retrograde 
motion — and all the observed irregularities are systematic in 
their irregularities, showing that they are governed, at least, 
by constant and invariable laws. If the earth is really sta- 
tionary, we cannot account for this retrograde motion of the 
planets, unless that motion is real; and if real, why, and 
how can it change from direct to stationary, and from station- 
ary to retrograde, and the reverse? 

Retrograde But if we conceive the earth in motion, and going the same 

motion ofthe wa y yfifo fa pi ane ^ an d moving more rapidly than the planet, 

planets ac- . 

counted for. t' ien l' ie 2 J ^ ane ^ Wl ^ cippear to run back ; that is, retrograde. 

And as this retrogradation takes place with every planet, 
when the earth and planet are both on the same side of the 
sun, and the planet in opposition to the sun ; and as these cir- 
cumstances take place in all positions from the sun, it is a suf- 
ficient explanation of these appearances ; and conversely, then, 
these appearances show the motion of the earth. 

(94.) When a planet appears stationary, it must be really 



PLANETARY MOTION. Il-j 

so, or be moving directly to or from the observer, And if it Cyxt vi. 
be moving to or from the observer, that circumstance will be Planets lev. 
indicated by the change in apparent diameter; and observa- erstationar y- 
tions confirm this, and show that no planet is really station- 
ary, although it may appear to be so. 

(95.) If we suppose the earth to be but one of a family of The earth a 
bodies, called planets — all circulating round the sun at clif- P lanet - 
ferent times — in the order of Mercury, Venus, Earth, Mars 
(omitting the small telescopic planets), Jupiter, Saturn, Her- 
schel, or Uranus, we can then give a rational and simple ac- 
count for every appearance observed, and without discussing 
the ancient objections to the true theory of the solar system, 
we shall adopt it at once, and thereby save time and labor, 
and introduce the reader into simplicity and truth. 

(96.) The true solar system, as now known and acknow- Copernicus 
ledged, is called the Cooernican system, from its discoverer, ani } e °" 

° x . pernican sys- 

Copernicus, a native of Prussia, who lived some time in the tem. 
fifteenth century. 

But this theory, simple and rational as it now appears, and Lost and re- 
capable of solving every difficulty, was not immediately adop- vived ' 
ted ; for men had always regarded the earth as the chief 
object in God's creation ; and consequently man, the lord of cre- 
tion, a most important being. But when the earth was hurled 
from its imaginary, dignified position, to a more humble 
nlace, it was feared that the dignity and vain pride of man 
must fall with it ; and it is probable that this was the root 
of the opposition to the theory. 

So violent was the opposition to this theory, and so odious Galileo and 
would any one have been who had dared to adopt it, that it his clialo s ue - 
appears to have been abandoned for more than one hundred 
years, and was revived by Galileo about the year 1620, who, 
to avoid persecution, presented his views under the garb of a 
dialogue between three fictitious persons, and the points left 
undecided. 

But the caution of Galileo was not sufficient, or his dia- 
logue was too convincing, for it woke up the sacred guardians 
of truth, and he was forced to sign a paper denouncing the 
theory as heresy, on the pain of perpetual imprisonment. 
8 



104 



ASTRONOMY. 



Chai-. VI. 



But this is a digression. With the history of astronomy, as 
interesting as it may be, we design to have little to do, and 
to proceed only with the science itself. 



CHAPTER VII. 



tis 



FIRST APPROXIMATIONS OF THE RELATIVE DISTANCES OF THE 
PLANETS FROM THE SUN. HOW THE RESULTS ARE OBTAINED. 

(97.) Being convinced of the truth of the Copernican 

system, the next step seems to be, to find the periodical times 

of the revolutions of the planets, and at least their relative 

distances from the sun. 

Distinction Mercury and Venus, never coming in opposition to the sun, 

between in- but revolving around that body in orbits that are within that 

penor plan- °^ *' ae ear th, are called inferior planets. 

Those that come in opposition, and thereby show that 
their orbits are outside of the earth, are called superior 
planets. 

We shall show how to investigate and determine the posi- 
tion of one inferior planet ; and the same principles will be 
sufficient to determine the position of any inferior planet. 

It will be sufficient, also, to investigate and determine the 
orbit of one superior planet; and if that is understood, it may 
be considered as substantially determining the orbits of all 
the superior planets ; and after that, it will be sufficient to 
state results. 

For materials to operate with, we give the following table 
of the planetary irregularities ( so called ) drawn from obser- 
vation : 



Planets. 



Mercury. 

Venus. 

Earth. 

Mars. 

Jupiter. 

Saturn. 

Uranus. 



Greatest 
Apparent 
Diameters. 



Least 
Apparent 
Diameters. 



(Angular Dist.j 
(from Sun at the 
(instant of being 
stationary. 



11.3 
59.6 

171 

44.5 

20.1 

4.1 



5.0 
9.6 

3.6 
301 
16.3 

3.7 



18 00 
28 48 

136 48 
115 12 
108 54 
103 30 



Mean arc of ' 
Retrogradation . 1 


° , 


13 30 
16 12 


16 12 
9 54 
6 18 
3 36 



PLANETARY MOTION. 



105 



Planets. 


Mean Duration of the Retro- 
grade Motion. 


Mean Duration of the Synodic 

Revolution, or interval between 

two successive oppositions. 


Mercury. 
Venus. 


23 days. 
42 " 


118 days. 

584 « 


Earth. 






Mars. 


73 " 


780 « 


Jupiter. 
Saturn. 


121 " 
139 " 


399 " 

378 « 


Uranus. 


151 " 


370 " 



Chap. VII. 



In the preceding table, the word mean is used at the head why the 
of several columns, because these elements are variable — word MEAW 

.. 'lii ii should be 

sometimes more and sometimes less, than the numbers here use d. 
given — which indicates that the planets do not revolve in cir- 
cles round the sun, but most probably in ellipses, like the orbit 
of the earth. 

On the supposition, however, that the planets revolve in 
circles ( which is not far from the truth ), the greatest and 
least apparent diameters furnish us with sufficient data to 
compute the distances of the planets from the sun in relation 
to the distance of the earth, taken as unity. 

(98.) In addition to the facts presented in the preceding The eionga- 
table, we must not fail to note the important element of the tionsofMer - 

„ i T-r m- • t t curyandVe- 

elongations ot Mercury and Venus. Ibis term can be applied nus 
to no other planets. 

It is very variable in regard to Mercury — showing that This element 
the orbit of that planet is quite elliptical. The variation is *J[jJ ble .. and 
much less in regard to Venus, showing that Yenus moves shows, 
round the sun more nearly in a circle. 

The least extreme elongation of Mercury is 

The greatest " " " is 

The mean (or the greatest elongation when 
both the earth and planet are at their 
mean distances from the sun ) is - 

The least extreme elongation of Venus is 

The greatest " " " is 

The mean (or at mean distances), is 

The least extremes must happen when the planet is in its 
perigee and the earth in its apogee, and the greatest when 
the earth is in perigee and the planet in apogee ; but it is 



37'. 



28° 4\ 



22° 46'. 
44° 58'. 
47° 30'. 
46° 30'. 



106 A3TRONOMY. 

Chv.>. vii. very seldom that these two circumstances take place at the 
same time. 
How to Belying on these facts as established by observations, we 

tin;] the com- .. i • • n -\r -re- 

parative can easily deduce the relative orbits of Mercury and Venus. 

magnitudes Let S (Fie. 23) re- 

of the orbits YlP. 23. , 

of Mercniy, ^ ^^j^ present the sun, i? the 

"encs, a,.a ^|j| J earth, V Venus. 

the earth SuSHfcsBrwi n • i i 

Conceive the planet 
to pass round the sun 
in the direction of A 
VJB. 

The earth, moves also 
in the same direction, 
but not so rapidly as 
Venus. 

Now it is clearly evi- 
dent, from inspection, 
that when the planet is 
passing by the earth, as 
at B, it will appear to 
pass along in the hea- 
vens in the direction of 
m to n. But when the planet is passing along in its orbit, at 
A, and the earth about the position of E, the planet will 
appear to pass in the direction of n to w. When the planet 
is at V, as represented in the figure, its absolute motion is 
nearly toward the earth, and, of course, its appearance is 
nearly stationary. 
What to j£ j g absolutely stationary only at one point, and even then 

understand , .. . 

by station- hut for a moment ; and that point is where its apparent mo- 
ar y- tion changes from direct to retrograde, and from retrograde 

to direct; which takes place when the angle SE V is about 
29 degrees on each side of the line SE. 

When the line E V touches the circumference A VJB, the 
angle S E J 7 , or cwjle of elongation, is then greatest ; and the 
triangle SE V\% right angled at V; and if SE is made ra- 
dius, S V will be the sine of the angle SE V 

But the line S E is assumed equal to unity, and then £ V 




PLANET A R Y MOTION. 107 

w ill be the natural sine of 46° 20'. and can be taken out of «<*.„ ™ 

L HAP, V 11, 

any table of natural sines ; or it can be computed by loga- 

rithms, and the result is .72336. 

For the planet Mercury, the mean of the same angle is 

22° 46'. and the natural sine of that angle, or the mean radius 

of the planet's orbit, is .38698. 

Thus we have found the relative mean distances of three 

planets from the sun, to stand as follows : 

Mercury, ------ 0.38698 

Venus, 0.72336 

Earth, 1.00000 

( 99. ^ If the orbits were perfect circles, then the angle The olbits 
b Jb V, oi greatest elongation, would always be the same; and y enES 
but it is an observed fact that it is not always the same; not circles. 
therefore the orbits are not circles ; and when S V is least, 
and S E greatest, then the angle of elongation is leant ; ami 
conversely, when S V is greatest and S E least, then the 
angle of elongation is the greatest possible ; and by observing 
in what parts of the heavens the greatest and least elongations 
take place, we can approximate to the positions of the longer 
axis of the orbits. 

( 100. ) By means of the apparent diameters, we can also Computa- 
find the approximate relations of their orbits. For instance, t . 10n ° 

I i ' rrom appa- 

when the planet Venus is at B, and appears on the sun's rent diame- 
disc, its apparent diameter is 59". 6 ; and when it is at A, or ters - 
as near A as can be seen by a telescope, its apparent diame- 
ter is 9";6: Now put 

SB=x; then EB=l—x, and AE=l-\-x. 

By Art. 66, 1— x : l+.i- : : 96 : 596; 

Hence, - - - - #=0.72254. 

By a like computation, the mean distance of Mercury from 
the sun, is 0.3864. 

(101.) To determine the mean relative distances of the 
superior planets from the sun, we proceed as follows : 

Let S (Fig. 24) represent the sun, E the earth, and i¥~one 
of the superior planets, say Mars. It is easy to decide, from 
observation, when the planet is in opposition to the sun. 



108 



ASTRONOMY, 



Chap. VII. 

Method of 
approximat- 
ing to the cr- 
bits of t! e 
superior pla- 



Fig. 24. This gives the position 

of S, E, and M, in one 
right line, in respect 
to longitude. Now by 
knowing the true angu- 
lar motion of the earth 
about the sun (73), and 
the mean angular mo- 
tion of the planet, * we 
can determine the angle 
mSe, corresponding to 
any definite future time ; 
for, by the motion of the 
earth round the sun, we 
can determine the angle 
E S e ; and by the mo- 
tion of the planet in the 
same time, we can determine the angle MS m ; and the dif~ 




The relative "gy means of apparent diameters, we can determine the 
values of the orbit. When the planet is in opposition to the 

p.anet from i a I 

the sun de- sun, at E (Fig. 24), measure its apparent diameter ; and, 
termmed oy a ft er a definite time, when the earth is at e, measure the ap- 

the v&na- . ' 

tion in its parent diameter again, and observe the angle S em. Pro- 
apparent dia. duce Se to n. Then, by the apparent diameters, we have 
the proportion of e m and e n (e n is the same as E M, brought 
to this position), and in the triangle emn we have the pro- 
portion between the two sides and the included angle men. 
These are sufficient data to determine the angles en wand 
emn', and their difference is the angle Sme. Now we can 
determine the side S m, of the triangle Sm e, and the triangle 
S em is completely known. Subtract the angle e Sm from 
the whole angle e S 31, and the angle M Sm is left. That 
is, while the earth is describing the angle E Se, the planet 
describes the angle M Sm. Put P for the periodical revo- 



* Here we anticipate a little ; for we have not shown how to deter- 
mine the periodical time of revolution from observation : but this is 
shown in a future chapter, and in the above text note 



PLANETARY MOTION. 109 

ference of these two angles is the angle m S e. By direct Chap. vrr. 
observation at e, we determine the angle Sem; and two 
angles, and the side S e, of the triangle $ m e, are sufficient to 
determine the side Sm, the value sought. The triangle 
gives the following proportion : 

. e 1 . e „ sin. Sem 

sin. bme : 1 :: sin. kern : &m=— — = — . 

sm. ome 

This is a general solution, for any superior planet ; but the why the 
result is only approximate ; for, until we know the eccentri- result is a ^ _ 
city of the orbit in question, and the part of the orbit in 
which the planet then is, we cannot accurately know the 
angle MSm. 

lution of the planet ; then, on the supposition of uniform 
motion, we have 

arc MSm : arc ESe : : 365i : P 

In this proportion the two arcs are known, and from thence 
P becomes known ; and thus, ive perceive, that by the variations 
of the apparent diameter of a planet, we can determine its rela- 
tive distance from the sun, and its periodical revolution. 

We give the following hypothetical example, for the pur- 
pose of further illustration. 

The apparent diameter of Mars, when in opposition to the sun, a problem 
tvas observed to be 17". 1. One hundred and eleven days after- 
ward, when the earth had passed over 110° of its orbit, the appa- 
rent diameter of Mars was again observed, and found to be 7 "A, 
and its angular position, in longitude, was 87° from the sun. 
From these data, it is required to find the relative approximate 
distance of the planet from the sun, and the approximate time of 
its revolution round the sun. 

From these data we have the angle M Sn=110°, Se m= its soiu 
87° ; therefore n e m=93°. tion - ~ Fi *- 

. 24. 

By the observed apparent diameter, we have EM to em 
as 7".4 to 17".l; but EM=en, therefore 

en : em : : 74 : 171. 
In the triangle nem we can take erc=74, and Em=171, 
for the purpose merely of finding the angles. Then, by trigo- 
nometry, we have 



110 



ASTRONOMY. 



C hap, vh . ( 102.) By a perusal of the last text note, it will be seen, 

Results by those even who are not expert mathematicians, that it is 

from vana- not diffi cu it to decide upon the relative distances of the 

tions in ap- ''."■'■. 

parent dia- planets from the sun, by observing their changes in apparent 
meters diameter, as seen from the earth. Such observations have 

been often made, and the following table shows the results; 

which are compared with the results deduced from Kepler's 

Third Law.* 



Planets. 


Deduced from appa- 
rent Diameters. 


From Kepler's 
Law. 


Difference or 
Error. 


Mercury . . . 
Jupiter 


0.386400 
0.722540 
1.000000 
1.533333 

5.180777 

9.579000 

19.500000 


0.387098 
0.723331 
1.000000 
1.523692 
5.202776 
9.538786 
19.182390 


—.000698 
—.000791 

+.009641 
— .021999 
+.040214 
+.317610 



Text note 
continued. 



171—74 



tan. 



tan. i, difference be- 



171+74 

—i 

tween the angle n and n m e. 

That is, - 245 : 97 : : tan. 43° 30' : tan. § Sme. 

Whence, >SW=41° 11'. Now in the triangle Sme, 
sin. 41° 11' : 1 : : sin. 87° : £m=1.517. 

Secondly, as the angle Sme=41 11' and Sem 87°, there- 
fore, - - m&=51 c 49', and MSm 58° 11'. 

But the times of revolution, between any two planets, must 
be inversely as the angles they describe in the same time ; 
the greater the angle, the shorter the periodic time; and 
therefore if we put P to represent the periodical revolution 
of Mars, we shall have 

58 T 2 F : 110 : : 365i : P. Hence P=690§ days. 

The true time is 686.97964; showing an error of a little 
more than three days ; but this is not a great error, consider- 
ing the remoteness of the data, and the want of minuteness and 
unity in the supposed observations. Our object is only to 
teach principles; not, as yet, to establish minute results. 



* A principle to be explained in Physical Astronomy. 



PLANETARY MOTION. Ill 

The distances drawn from Kepler's law, are considered chap, vii. 
more accurate than conclusions drawn from most other con- Why the 
siderations ; and it is rather remarkable that these deduc- results from 
tions from the apparent diameters agree as well as they do, ^f e " e e r " ca ,|' 
owing to the difficulty of settling the exact apparent diam- not be relied 
eter, by observation. Take the apparent diameter of Ura- upon for at: " 

' •* * * curacy. 

nus, for example, 3". 7 and 4".l and change either of them 
j 1 ^ of a second, and it will make a great difference in the 
deduced result. 



CHAPTER VIII. 

METHODS OF OBSERVING THE PERIODICAL REVOLUTIONS OF THE 
PLANETS, AND THEIR RELATIVE DISTANCES FROM THE SUN. 

( 103.) The subject of this chapter will be to explain the Chap - vin - 
principles of finding the periodical revolutions of the planets why direct 
around the sun. If observers on the earth were at the ° ,sena 10 " i 

are not to tae 

center of motion, they could determine the times of revo- point. 
lution by simple observation. But as the earth is one of the 
planets, and all observers on its surface are carried with it, 
the observations here made must be subjected to mathemati- 
cal corrections, to obtain true results ; and this was an impos- 
sible problem to the ancients, as long as they contended for a 
stationary earth. 

If the observer could view the planets from the center of Two impor* 
the sun, he would see them in their true places anions the tant p0Sl " 

1 ° tions. 

stars — and there are only two positions in which an observer 
on the earth will see a planet in the same place as though he 
viewed it from the center of the sun, and these positions are 
conjunction and opposition. 

Thus, in Fig. 24, when the earth is at E, and a planet at 
M, the planet is in opposition to the sun ; and it is seen pro- 
jected among the stars at the same point, whether viewed 
from S or from E. 

In Fig, 23, if the planet is at B, or A, it is said to be in C a a Xtbeob! 
conjunction with the sun; but a conjunction cannot be ob- served. 



112 



a& 1 RON MY: 



Chap. VIST. 



Revolution 
of inferior 
planets less, 
and of supe- 
ijor planets 
greater than 
a year. 



Times of 
opposition 
can. be ob- 
served 



served on account of the brilliancy of the sun, unless it be the 
two planets, Mercury and Venus, and then only when they 
pass directly before the face of the sun, and are projected on 
its surface as a black spot. Such conjunctions are called transits. 
( 104.) All the planets move around the sun in the same 
direction, and not far from the «ame plane, and the rudest 
and most careless observations show that those planets near- 
est the sun, perform their revolutions in shorter periods than 
those more remote. From this, we decide at once that the 
mean angular motion of all the superior planets is less than 
the mean angular motion of the earth in its orbit ; and the 
mean angular motion of the inferior planets, as seen from 
the sun, is greater than the mean motion of the earth. 

(105.) The time that any planet comes in opposition to 
the sun, can be very distinctly determined by observation. 
Its longitude is then 180 degrees from the longitude of the 
sun, and comes to the meridian nearly or exactly at midnight. 
If it is a little short of opposition at the time of one obser- 
vation, and a little past at another, the observer can propor- 
tion to the exact time of opposition, and such time can be 
definitely recorded — and by such observation, we have the 
true position of the planet, as seen from the sun. Another 

opposition of the same kind and 
of the same planet, can be ob- 
served and recorded. 

The elapsed time between two 
such oppositions is called the sy- 
nodical revolution of the planet. 
We note the time that a 
planet is in opposition to the 
sun. Then S E and M are in 
one plane as represented in Fig. 
25. If the planet M should 
remain at rest while the earth, 
E made its revolution; then 
the synodical revolution would 
be the same as the length of 
our year. But all the planets move in the same direction as 





Fig 
S 


\ 25. 


Synodical 






revolution. 






Mean angu- 
lar motion of 
the planets \^ 
determined 




E — \^* 


from their 

synodical 

revolutions. 




Itv 



PLANETARY MOTION. 113 

the earth; and therefore the earth, after making a revolu- chap. vm. 
tion, must pass onward and employ additional time to over- 
take the planet ; and the more rapidly the planet moves, the 
longer time it will require. Hence, in case two planets have 
but a small difference in angular motion, their synodical pe- General con- 
ned must be proportionately long. The planet Jupiter sicterations. 
moves about 31° in its orbit in a year; and therefore, after 
one opposition, the earth is round to the same point in 365| 
days, and to gain the 31° requires about 32 days more ; hence 
the synodical revolution of Jupiter must be about 397 days, 
by this very rough and imperfect computation. By inspect- 
ing the table on page 105, we perceive that the mean synodi- 
cal revolution of Jupiter is 399 days, and this observed fact 
shows us that Jupiter passes over about 31° in a year, and of 
course its revolution must be a little less than 12 years; and 
by the same considerations, we can form a rough estimate of 
the periodical revolutions of all the planets. 

( 106.) The general principle being understood, we may 
now be more scientific. The mean motion of the earth Computation 
in its orbit is very accurately known. Represent its daily I? determine 

•> •> J- •> the mean an- 

motion by a. The angular motion of the planet ( any supe- guiai motion 
rior planet that maybe under consideration) is unknown ; of the earth * 
therefore, represent its daily motion by x. Let the angle F 
S e represent a, and the angle M S m represent x ; then the 
angle m Se or ( a — x ) will represent the daily angular advance 
of the earth over the planet ; and as many times as the an- 
gle m S e is contained in 360° will be the number of days in 

Q f\C\ 

a synodical revolution. Therefore, = the observed 

a — x 

time of a synodical revolution ; and by taking the times from 
the table (page 105), we have the following equations : 

Mars. Jupiter. Saturn. Uranus. 

360 :7 80, J«L=899, J^=378, -^-=370.* 



a — x a — x a — x a — x 

d 

* These equations correspond to the general equation f= in 

c— x 
Robinson's Algebra, page 105, University edition. 

8 J* 



114 ASTRONOMY. 

Chap. viii. The value of a is 59' 8", and then a solution of these sev- 
eral equations gives the mean angular motion, per day, of the 
several planets, as follows : 

Mars. Jupiter. Saturn. Uranus. 

31' 27" 4' 59".4 V 59".5 45".3 

Times of Dividing the whole circle 360° by the mean daily motion 
aeXed°fr m °^ ea °k P^ ane *» w ^ §i ye their respective times of revolution, 
the angular and the following are the results : 

Mars, Jupiter. Saturn. Uranus. 

687 days. 4331 days. 10840 days. 28610 days. 
( 106.) For the inferior planets, Mercury and Venus, we 
have the same principle, only making x greater than a, and 

J£~p For Mercury. For Venus. #$- 

!?i-118 ; -^=584. 

x — a x — a 

x=A°2' 11"; x=l° 36' 7". 

Mean an- These diurnal angular motions correspond to 89 days for 
guiar motion tlie revolution Q f Mercury, and 224.8 days for the revolution 

of the inferior J * 

planets, and of Venus. All these results are, of course, understood as 
their revoiu- g rs £ approximations, and accuracy here is not attempted. 

tion round x ... . . 

the sun. We are only showing principles ; and it will be noticed, that 
the times here taken in these considerations, are only to the 
nearest days ; and not fractions of a day, as would be necessary 
for accurate results. By this method accuracy is never at- 
tempted, on account of the eccentricity of the orbits. No 
two synodical revolutions are exactly alike ; and therefore 
it is very difficult to decide what the real mean values are. 

( 107.) To obtain accuracy, in astronomy, observations 
must be carried through a long series of years. The follow- 
ing is an example; and it will explain how accuracy can be 
attained in relation to any other planet. 

On the 7th of November, 1631, M. Cassini observed Mer- 
cury passing over the sun ; and from his observations then 
taken, deduced the time of conjunction to be at 7 h. 50 m., mean 
time, at Paris, and the true longitude of Mercury 44° 41' 35". 
observa- Comparing this occultation with that which took place in 

tions carried 1723, the true time of conjunction was November 9th, at 5 h. 

ion- "course 29m., p. m., and Mercury's longitude was 46° 47' 20". 



PLANETARY MOTION. 115 

The elapsed time was 92 years, 2 days, 9 h. 39 m. Twenty- chap. vin. 
two of these years were bissextile ; therefore the elapsed time of t0 

was (92x365) days, plus 24 d. 9h. 39 m. secure accu- 

In this interval, Mercury made 382 revolutions, and 2° 5' rdCy ' 
45" over. That is, in 33604.402 days, Mercury described 
137522.095826 degrees; and therefore, by division, we find 
that in one day it would describe 4°.0923, at a mean rate. 

Thus, knowing the mean daily rate to great accuracy, the 
mean revolution, in time, must be expressed by the fraction 

Q9 ; or, 87.9701 days, or 87 days 23 h. 15 m. 57 s. 

( 108. ) The following is another method of observing the Another 
periodical times of the planets, to which we call the student's |, servn Z 
special attention. periodical re- 

The orbits of all the planets are a little inclined to the volBtions of 

..-•-. the planets. 

plane of the ecliptic. 

The planes of all the planetary orbits pass through the 
center of the sun ; the plane of the ecliptic is one of them, 
and therefore the plane of the ecliptic and the plane of any 
other planet must intersect each other by some line passing- 
through the center of the sun. The intersection of 'two planes 
is always a straight line. (See Geometry.) 

The reader must also recognize and acknowledge the fol- 
lowing principle : 

That a body cannot appear to be in the plane of an observer, 
unless it really is in that plane. 

For example ; an observer is always in the plane of his 
meridian, and no body can appear to be in that plane unless 
it really is in that plane; it cannot be projected in or out of 
that plane, by parallax or refraction. 

Hence, when any one of the planets appears to be in the 
plane of the ecliptic, it actually is in that plane; and let the 
time be recorded when such a thing takes place. 

The planet will immediately pass out of the plane, because What is 
the two planes do not coincide. Passing the plane of the meant b ^ 

. . node. 

ecliptic is called passing the node. Keep track of the planet 
until it comes into the same plane ; that is, crosses the other 
node ; in this interval of time the planet has described just 



116 ASTRONOMY. 

Chap. tiii. 180°, as seen from the sun (unless the nodes themselves are 
Two nodes in motion, which in fact they are ; but such motion is not 
180 degrees sensible for one or two revolutions of Venus or Mars). 
^-hlr,c= Q I„ Continue observations on the same planet, until it comes 
from the sun. into the ecliptic the second time after the first observation, 
or to the same node again, and the time elapsed, is the time of 
a revolution of that planet round the sun. From such observa- 
tions the periodical time of Venus became well known to 
astronomers, long before they had opportunities to decide it 
by comparing its transits across the sun's disc ; and by thus 
knowing its periodical time and motion, they were enabled to 
calculate the times and circumstances of the transits which 
happened in 1761, and in 1769; save those resulting from 
parallax alone. 
First idea of (109.) On comparing the time that a planet remains on 
of the plan- eacn s ^ e °f ^he Ecliptic, we can form some idea of the position 
ets. of its apogee and perigee. If it is observed to be on each side 

of the ecliptic the same length of time, then it is evident that 
the orbit of the planet is circular, or that its longer axis coin- 
cides with its nodes. If it is observed to be a shorter time 
north of the plane of the ecliptic than south of it, then it is 
evident that its perigee is north of the ecliptic; but nothing 
more definite can be drawn from this circumstance. 
Finairesuits. ( HO.) Finally. By the combination of the different 
methods, explained in articles ( 98 ), ( 100 ), ( 101 ), ( 105 ), 
(107 ), and (108), and extending the observations through 
a long course of years, and from age to age, the times of rev- 
olution, the mean relative distances of the planets from the 
sun, were approximated to, step by step, until a great degree 
of exactness was attained, and the following were the results : 

Sidereal Revolution. Mean distance from Q. 

Mercury, - - - 87.969258 0.387098 

Venus, - - - 224.700787 0.723332 

Earth, - - - 365.256383 1.000000 

Mars, - - - 686.979646 1.523692 

Jupiter, - - 4332.584821 5.202776 

Saturn, - - 10759.219817 9.538786 

Uranus, - - 30686.820830 19.182390 



PLANETARY MOTION. 117 

( 111.) By inspecting the preceding table, we find that the Cha?. viil 
greater the distance from the- snn, the greater the time of Times one y- 
revolution ; but the ratio for the time is greater than the ratio olut:on ail;1 

-,'. distances 

corresponding to distance ; yet we cannot doubt that some 0O mpired 
connection exists between these ratios. 

For instance, let us compare the Earth with Jupiter. The 
ratio between their times of revolution, is near 12. 

The ratio between their relative distances from the sun, as 
we perceive, is nearly 5.2. 

The square of 12 is 144; the cube of 5.2 is near 141. 
But 12 is a little greater than the real ratio between the 
times of revolution, and 5.2 is not quite large enough for the 
ratio of distance, and by taking the correct ratios, they seem 
to bear the relation of square to cube. 

Without a very rigid or close examination, we perceive 
that five revolutions of Jupiter are nearly equal to two revolu- 
tions of Saturn; that is, f is nearly the ratio between their 
times of revolution. 

By inspecting the column of distances, we perceive that 
the ratio of the distances of these two planets, is nearly |f ; 
and if we square the first ratio, and cube the second, we shall 
have nearly the same ratio. 

Now let us compare two other planets, say Venus and Result dts- 
Mars, more exactly. 

Their ratio of revolution is 686,979 log. - 2.836948 

224,701 log- - 2.351601 

Log. of the ratio, - - - 0.485347 

Multiply by 2 

Log. of the square of the ratio of time, 0.970694 

Their ratio of distance is, 15.23692 log. - 1.182883 

7.23332 log. - 859323 

Log. of the ratio, - - - 0.323560 

Multiply by - 3 

Log. of the ache of the ratio of distance, 0.970680 
Thus we perceive that the squares of the times of revolu- 
tion, are to each other as the cubes of the mean distances of 



covered. 



118 ASTRONOMY 

Chap. viii. the planets from the sun,* and this is called Kepler's third 
Kepler's low ; and it was by such numerical comparisons that Kepler 
iaws " discovered the law.f 

We may now recapitulate the three laws of the solar sys- 
tem, called Kepler's laws, as they were discovered by that 
philosopher. 

1st. The orbits of the planets are ellipses, of which the sun 
occupies one of the foci. 

2d. The radius vector in each case, describes areas about the 
focus, vjhich are proportional to the times. 

2d. The square of the times of revolution, are to each other 
as the cubes of the mean distances from the sun. 



* For a concise mathematical view of this subject, we give 
the following: Let d and D represent mean distances from 
the sun, and t and T the times of revolution. Then 

T D ; 

— -== n, ~j— m > n an d m taken to represent the ratios. 

Square the 1st equation and cube the 2d. Then 

— -=n 2 , and — — =m 3 
t 2 ' d 3 

But by inspection we know that 

T 2 D 3 

n 2 =m 3 : therefore, — = -— , or, t 2 : T 2 ::d 8 : D 3 . 
' ' t 2 d 3 

j It appears that Kepler did not compare ratios, as we have done ; 
but took the more ponderous method of comparing the elements of the 
ratios (the numbers themselves ) ; for, says the historian : — It was on 
the 8th of March, 1618, that it first came into Kepler's mind to com- 
pare the powers of the numbers which express their revolutions and 
distances ; and by chance he compared the squares of the times with 
the cubes of the distances ; but from too great anxiety and impa- 
tience, he made such errors in computation, that he rejected the hy- 
pothesis as false and useless ; but on examining almost every other 
relation in vain, he returned to the same hypothesis, and on the 15th 
of May, of the same year, he renewed his calculation with completo 
success, and established this law, which has rendered his name im- 
mortal 



SOLAR PARALLAX. H9 



CHAPTER IX. 

TRANSITS OP VENUS AND MERCURY. HOW SUN'S HORIZONTAL 

PARALLAX DEDUCED 

(112.) We have thus far been very patient in our inves- chap, ix. 
tigations — groping along — finding the form of the planetary Alttempts to 
orbits, and their relative magnitudes ; but, as yet, we know find the sun's 
nothing of the distance to the sun; save the indefinite fact, P arallax - 
that it must be very great, and its magnitude great; but 
how great we can never know, without the sun's parallax. 
Hence, to obtain this element, has always been an interesting 
problem to astronomers. 

The ancient astronomers had no instruments sufficiently * ; jj;ffi cu it; es 
refined to determine this parallax by direct observation, in the of ancient 
manner of finding that of the moon (Art. 60), and hence the astronomers - 
ingenuity of men was called into exercise to find some artifice 
to obtain the desired result. 

After Kepler's laws were established, and the relative dis- 
tances of the planets made known, it was apparent that their 
real distance could be deduced, provided the distance between 
the earth and any planet could be made known. 

(113.) The relative distances of the earth and Mars, from Parallaxof 
the sun (as determined by Kepler's law) are asl to 1.5237; Mars. 
and hence it follows that Mars, in its oppositions to the sun, 
is but about one half as far from the earth as the sun is ; and 
therefore its parallax (Art. 60) must be about double that 
of the sun; and several partially successful attempts were 
made to obtain it by observation. 

On the 15th of August, 1719, Mars being very near its Maraidi 
opposition to the sun, and very near a star of the 5th mag- approx^ma^ 
nitude, its parallax became sensible ; and Mr. Maraldi, an tion to the 
Italian astronomer, pronounced it to be 27". The relative pj™ 31 " oi 
distance of Mars, at that time, was 1.37, as determined from 
its position and the eccentricity of its orbit. 

But horizontal parallax is the angle under which the earth 
appears ; and, at a greater distance, it will appear under a 
9 



120 ASTRONOMY. 

Chap. ix. less angle. The distance of Mars from the earth, at that 
time, was .37, and the distance of the sun was 1 ; therefore, 
1 : .37 :: 27" : 9".99, or 10", nearly, for the sun's horizon- 
tal parallax. 
o>erva- On the 6th of October, 1751, Mars was attentively ob- 

eentia and serve( ^ D y Wargentin and Lacaille (it being near its opposi- 

Lacaiiie tion to the sun), and they found its parallax to be 24" .6, 
from which they deduced the mean parallax of the sun, 10".7. 
But at that time, if not at present, the parallax of Mars 
could not be observed directly, with sufficient accuracy to 
satisfy astronomers ; for no observer could rely on an angu- 
lar measure within 2" ; for full that space was eclipsed by 
the micrometer wire. 
Dr. Hal- (114.) Not being satisfied with these results, Dr. Halley, 

ey s sugges- aQ jjjjgjjg^ astronomer, very happily conceived the idea of 
finding the sun's parallax by the comparisons of observa- 
tions made from different parts of the earth, on a transit of 
Venus over the sun's disc. If the plane of the orbit of Venus 
coincided with the orbit of the earth, then Venus would come 
between the earth and sun, at every inferior conjunction, at 
intervals of 584.04 days. But the orbit of Venus is inclined 
to the orbit of the earth by an angle of 3° 23' 28" ; and, in 
the year 1800, the planet crossed the ecliptic from south to 
north, in longitude 74° 54' 12", and from north to south, in 
longitude 254° 54' 12": the first mentioned point is called 
The nodes the ascending node ; the last, the descending node. The nodes 

of venns. retrograde 3r 10 " i n a century. 

What times (115.) The mean synodical revolution of 584 days corre- 
in the year S p 0ndg ^jj n0 aliquot part of a year; and therefore, in the 

tiansiis may x . i • 

take place, course of time, these conjunctions will happen at different 
points along the ecliptic. The sun is that part of the ecliptic 
near the nodes of Venus, June 5th and December 6th or 7th ; 
and the two last transits happened in 1761 and in 1769 ; and 
from these periods we date our knowledge of the solar parallax. 
Revoiu- (H6.) The periodical revolution of the earth is 365.256383 

tions com- d and tnat of y enus ig 224.700787 ; and as numbers they 

pared. . . 

are nearly m proportion of 13 to 8. 

From this it follows, that eight revolutions of the earth 



SOLAR PARALLAX. 121 

require nearly the same time as r 13 revolutions of Venus; Chap. ix 
and, of course, whenever a conjunction takes place, eight 
years afterward another conjunction will take place very near 
the same point in the ecliptic* 



* The ratio of the times of these revolutions is directly Compara. 

; , . , 224.700787 , . . ei ;° m ° tion * 
compared, as terms of a traction, thus, ._. ^^r,-. '■> an « ^ 1S venusar,a 

r 355.256381 the earth. 

manifest that 365.256383 days, multiplied by the number 
224700787, will give the same product as 224.700787 days 
multiplied by the number 365256383 ; that is, after an elapse 
of 224700787 years, the conjunction will take place at the 
same point in the heavens; and all intermediate conjunctions 
will be but approximations to the same point : and to obtain 
these approximate intervals, we reduce the above fraction to 
its approximating fractions, by the principle of continued 

fractions. f See Robinson's Arithmetic. ) 

The approximating fractions are 

112 3 8 235 
I' 2' 3' 5' 13' 382' 

To say nothing of the first two terms, these fractions show 
that two revolutions of the earth are near, in length of time, 
to three revolutions of Venus ; three revolutions of the earth 
a nearer value to five revolutions of Venus : and eight revo- 
lutions of the earth a still nearer value to 13 revolutions of 
Venus ; and 235 revolutions of the earth a very near value 
to 382 revolutions of Venus. 

The period of eight years, under favorable circumstances, 
will bring a second transit at the same node ; but if not in 
eight years, it will be 235 years, or 235-f-8=243 years. 

For a transit at the other node, we must take a period of 
235 — 8 years, divided by 2, or 113 years; and sometimes 
the period will be eight years less than this, or 105 years. 
The first transit known to have been observed was in 1639, 
December 4th; to this add 235 years, and we have the time 
of the next transit, at the same node, 1874, December 8th; 
and eight years after that will be another, 1882, December 
6th. The first transit observed at the ascending node, was 

K 



122 ASTRONOMY. 

Chap. ix. If the proportion had been exactly as 13 to 8, then the 

Periods of conjunctions would always take place exactly at the same 

conjunctions p 0m t . fo^, as it is, the points of conjunction in the heavens 

at the same . ,. • 

time of the are east and west of a given point, and approximate nearer 

year. an( j nearer to that point as the periods are greater and 

greater. 

Only two To be more practical, however, the intervals between con- 

can i unctions are such, combined with a slight motion of the nodes, 

happen at in- ° ' % * 

tervais of 8 that the geocentric latitude of Venus, at inferior conjunctions 
years. near t ne ascending node, changes about 19' 30" to the north, 

in the period of about eight years. At the descending node, 
it changes about the same quantity to the southward, in the 
same period ; and as the disc of the sun is but little over 32', 
it is impossible that a third transit should happen 16 years 
after the first; hence only two transits can happen, at the 
same node, separated by the short interval of eight years. 
Periods be- (117.) If at any transit we suppose Venus to pass directly 

transits of over *^e center °f * ne sun > as seen from the center of the 
Venus. earth — that is, pass conjunction and node at the same time — 
at the end of another period of about eight years, Venus 
would be 19' 30" north or south of the sun's center; but as 
the semidiameter of the sun is but about 16', no transit could 
happen in such a case ; and there would be but one transit 
at that node until after the expiration of a long period of 235 
or 243 years. 

After passing the period of eight years, we take a lapse of 
105 or 113 years, or thereabouts, to look for a transit at the 
other node. 
Transits ( 118. ) Knowing the relative distances of Venus, and the 

can be com- in i i •• -, ... 

puted. earth, from the sun — the positions and eccentricities ot both 

Dr. Haiiey orbits — also their angular motions and periodical revolutions — 
to°find the ever y circumstance attending a transit, as seen from the 
sun's parai- earth's center, can be calculated; and Dr. Halley, in 1677, 
lax ' read a paper before the London Astronomical Society, in 

Text note [ n 17(31 j une 5th ; eight years after, 1769, June 3d, there 
was another ; and the next that will occur, at that node, will 
be in 2004, June 7th, 235 years after, 1769. 



SOLAR PARALLAX. 123 

vvhich he explained the manner of deducing the parallax of Chap. ix. 
the sun, from observations taken on a transit of Yenus or 
Mercury across the sun's disc, compared with computations 
made for the earth's center, or by comparing observations 
made on the earth, at great distances from each other. 

The transits of Yenus are much better, for this purpose, Why the 
than those of Mercurv ; as Yenus is larger, and nearer the * ransits of 

* ° Venus are 

earth, and its parallax at such times much greater than that better adapt- 
of Mercury; and so important did it appear, to the learned ed t0 £ ive 
world, to have correct observations on the last transit of rallax than 
Yenus, in 1769, at remote stations, that the British, French, those of Mer - 
and Russian governments were induced to send out expedi- cur3 ' 
tions to various parts of the globe, to observe it. " The fa- 
mous expedition of Captain Cook, to Otaheite, was one of 
them." 

(119.) The mean result, of all the observations made on The result 
that memorable occasion, gave the sun's parallax, on the day 
of the transit (3d of June), 8". 5776. The horizontal paral- 
lax, at mean distance, may be taken at 8". 6 ; which places 
the sun, at its mean distance, no less than 23984 times the 
length of the earth's semidiameter, or about 95 millions of 
miles. 

This problem of the sun's horizontal parallax, as deduced The impor- 
from observations on a transit of Yenus, we regard as the tance of this 
most important, for a student to understand, of any in astro- 
nomy ; for without it, the dimensions of the solar system, and 
the magnitudes of the heavenly bodies, must be taken wholly 
on trust; and we have often protested against mere facts 
being taken for knowledge. 

(120.) We shall now attempt to explain this whole matter A general 
on general principles, avoiding all the little minutiae, which ex P lanatl0n - 
render the subject intricate and tedious; for our only object 
is to give a clear idea of the nature and philosophy of the 
problem. 

Let S (Fig. 26) represent the sun, and m n and P Q small 
portions of the orbits of Yenus and the earth. 

As these two bodies move the same way, and nearly in the 
same plane, we may suppose the earth stationary, and Yenus 



124 



ASTRONOMY. 



Chap. IX. 

The case 
simplified. 



An abstract 
proposition 
for the pur- 
pose of illus- 
tration. 



to move with an angular velocity, 
equal to the difference of the two. 

When the planet arrives at v, an 
observer at A would see the planet 
projected on the sun, making a dent 
at v . 

But an observer at G would not 
see the same thing until after the 
planet had passed over the small are 
v q, with a velocity equal to the diff- 
erence between the angular motion 
of the two bodies; and as this will 
require quite an interval of absolute 
time, it can be detected; and it mea- 
sures the angle A v' G; an angle 
under which a definite portion of the 
earth appears as seen from the sun. 

(121.) To have a more definite 
idea of the practicability of this me- 
thod, let us suppose the parallactic 
angle, A v' G, equal to 10", and in- 
quire how long Yenus would be in 
passing the relative arc v q. 

1° 36' 8" in a day. 
59' 8" " 

The relative, or excess motion of Venus for a mean solar 
day is then 37'. 

Now, as 37'' is to 24h. so is 10" to a fourth term; or, as 
2220" : 1440m. :: 10" : 6 m. 29 s. 

Now if observation gave more than 6 minutes and 29 sec- 
onds, we shall conclude that the parallactic angle was more 
than 10"; if less, less. But this is an abstract proposition. 
When treating of an actual case in place of the mean motion, 
we must take the actual angular motions of the earth and 
Venus, at that time, and we must know the actual position of 
the observers ; A and G, in respect to each other, and the po- 
sition of each in relation to a line joining the center of the 




Venus, at its mean rate, passes - 
The earth, 



SOLAR PARALLAX. 125 

earth and the center of the sun ; and then by comparing the Chap. ix. 
local time of observation made at A, with the time at G, and 
referring both to one and the same meridian, and we have the 
interval of time occupied by the planet in passing from v to 
q, from which we deduce the parallactic angle A v' G, and 
from thence the horizontal parallax. 

The same observations can be made when the planet passes A combiua- 
off the sun, and a great many stations can be compared with b ti ^3 
A, as well as the station G. In this way, the mean result of 
a great many stations was found in 1761, and in 1769, and 
the mean of all cannot materially differ from the truth. 

( 122.) There is another method of considering this whole Another me- 
subject, which is in some respects more simple and preferable thodofdedu - 
to the one just explained. It is for the observers at every b]eni . 
station to keep the track of the transit all the way across the 
sun's disc, and take every precaution to measure the length 
of chord upon the disc, which can be done by carefully noting 
the times of external and internal contacts, and the begin- 
ning and end of the transit, and at short intervals carefully 
measuring the distance of the planet to the nearest edge of 
the sun by a micrometer. 

If the parallax is sensible, it is evident that two observers, situation cf 
situated in different hemispheres, will not obtain the same dlfferent ob - 

x m servers. 

chord. For example, an observer in the northern hemisphere, 
as in Sweden or Norway, will see Venus traversing a more 
southern chord than an observer in the southern hemisphere. 
Now if each observer gives us the length of the chord as ob- 
served by himself, and, knowing the angular diameter of the 
sun, we can compute the distance of each chord from the 
sun's center, and of course we then have the angular breadth 
of the zone on the sun's disc between them. But as this 
zone is formed by straight lines passing through the same 
point, the center of Venus, its absolute breadth will depend on 
its distance from the point v; that is, the two triangles ABv 
and a b v ( Fig. 27) will be proportional, and we have 

A v : a v : : A B : a b. The result 

But the first three of these terms are known ; therefore the 
fourth, a b, is known also ; and if any definite angular space 

K* 



126 



ASTRONOMY. 



Chap. IX. Fig. 27. 



Under what 
circumstan- 
ces this me- 
thod should 
not be used. 



Transits of 
Mercury not 
important. 



Revolutions 
of Mercury 
and the earth 
compared. 




on the sun becomes known, the whole sem- 
idiameter becomes known, and from thence 
the horizontal parallax is immediately dedu- 
ced.* 

(123.) The accuracy of this method should be 
questioned when Venus passes near the sun's 
center, for the two chords are never more than 
30" asunder, and hence they will not percepti- 
bly differ in length when passing near the sun's 
center, and Venus will be upon the sun nearly 
the same length of time to all observers. 

( 124.) The apparent diameter of Mercury 
and Venus can be very accurately measured 
when passing the sun's disc. In 1769 the di- 
ameter of Venus was observed to be 59". 

( 125.) The same general principles apply 
to the transits of Mercury and Venus ; but those 
of Mercury are not important, on account of the 
smaller parallax and smaller size of that planet ; 
but owing to the more rapid revolution of Mer- 
cury, its transits occur more frequently. The 
frequent appearance of this planet on the face 
of the sun, gives to astronomers fine opportu- 
nities to determine the position of its node and 
the inclination of its orbit. 
In 1779, M. Delambre, from observations on the transit of 
May 7, placed the ascending node, as seen from the sun, in 
longitude 45° 57' 3". From the transit of the 8th of May, 
1845, as observed at Cincinnati, it must have been in longi- 
tude 46° 31' 10"; this gives it a progressive motion of about 
1° 10' in a century. The inclination of the orbit is 7° 0' 13". 
The periodical time of revolution is 87.96925 days ; that of 
the earth is 365.25638 days, and by making a fraction of 
these numbers, and reducing as in the last text note, we find 



LB 



* That is, as the real diameter of the sun, is to the real diameter of 
the earth, so is the sun's angular semidiameter to its horizontal par- 
allax. ( See 66). 



PLANETARY PARALLAX. 127 

that 6, 7, 13, 33, 46, 79, and 520 years, or revolutions of the Chap, ix. 
earth nearly correspond to complete revolutions of Mercury. 
Hence we may look for a transit in 6, 7, 13, 33, 46, &c, 
years, or at the expiration of any combination of these years 
after any transit has been observed to take place ; and by 
examining the following table, the years will be found to fol- Intervals be - 

° . . tween tran- 

low each other by some combination of these numbers. S i ts . 

The following is a list of all the transits of Mercury that 
have occurred, or will occur, between the years 1800 and 
1900: 

At the ascending node. At the descending node. 



1802, - 


- - Nov. 8. 


1799, - - - May 7. 


1822, 


- - - Nov. 4. 


1832, - - - May 5. 


1835, - 


- - Nov. 7. 


1845, - - - May 8. 


1848, 


- - - Nov. 9. 


1878, - - -May 6. 


1861, - 


- - Nov. 11. 


1891, - - - May 9. 


1868, 


- - - Nov. 4. 




1881, - 


- - Nov. 7. 




1894, 


- - - Nov. 10. 





CHAPTER X. 

THE HORIZONTAL PARALLAXES OF THE PLANETS COMPUTED, AND 
FROM THENCE THEIR REAL DIAMETERS AND MAGNITUDES. 

( 126.) Having found the real distance to the sun, acd the Chap. x. 
sun's horizontal parallax, we have now sufficient data to find Real mag. 
the real distance, diameter, and magnitude, of every planet nitudes ancl 

distsncGS c?-n 

in the solar system. now be de . 

In Art. 60 we have explained, or rather defined, the hori- termined 
zontal parallax of any body to be the angle under which the 
semidiameter of the earth appears, as seen from that body ; 
and if the earth were as large as the body, the apparent diame- 
ter of the body, and its horizontal parallax, would have the 
same value. And, in general, the diameter of the earth is to 
the diameter of any other planetary body, as the horizontal 
parallax of that body is to its apparent semidiameter. 

The mean horizontal parallax of the sun, as determined in 



128 ASTRONOMY. 

Chap. x. the last chapter, is 8".6; the semidiameter of the sun, at the 
Real dia- corresponding mean distance, is 16' 1", or 961". Now let d 
meter of the represent the real diameter of the earth, and D that of the 
Mm e er- g ^ ^ en we g^rj have -^he following proportion : 

d : D : : 8".6 : 961".0. 

But d is 7912 miles; and the ratio of the last two. terms is 

111.66; therefore D=(111.66)(7912)=883454 miles. 

Real dis- ( 127.) The sun's horizontal parallax is the angle at the 

tance be- k ase f a right angled triangle ; and the side opposite to it is 

earth and sun * Qe ra< &us of the earth (which, for the sake of convenience, 

determined, we now call unity). Let x represent the radius of the earth's 

orbit; then, by trigonometry, 

sin. 8".6 : 1 :: sin. 90° : x\ 

sin 90° 
Therefore, ^=-r— ^-=log. 10.00000— -log. 5.620073 * 
sm.8".6 to fe 

That is, the log. of #=4.379927, or z=23984 ; which is 
the distance between the earth and sun, when the semidia- 
meter of the earth is taken for the unit of measure ; but, for 
general reference, and to aid the memory, we may say the 
distance is 24000 times the earth's semidiameter. 

(128.) Now let us change the unit from the semidiameter 
of the earth to an English mile ; and then the distance be- 
tween the earth and sun is 
Bistancein (3956)(23984)=94880706 ; 

round num- • \ ■ 

!)er3- and, in round numbers, we say 95 millions of miles. 

By Kepler's third law, we know the relative distances of 

* Students generally would be unable to find the sine of 8". 6, or the 
sine of any other very small arc ; for the directions given in common 
works of trigonometry are too gross, and, indeed, inaccurate, to meet 
the demands of astronomy. 

On the principle that the sines of small arcs vary as the arcs them- 
selves, we can find the sine of any small arc as follows : 

Sine of 1', taken from the tables, is - - - - 6.463726 

Divide by 60, that is, subtract the log. of 60, - - 1. 778151 

The sine'of 1", therefore, is - - - - - 4. 685575 
Multiply by the number 8.6 ; that is, add log. - 0. 934498 

The sine of 8". 6, therefore, must be, - - - - 5. 620073 
In the same manner, find the sine of any other small arc. 



PLANETARY PARALLAX. 129 

all the planets from the sun; and now, having found the real Chap, x. 
distance of the earth, we may have the distance in miles, by How to 
multiplying the distance of the earth by the ratio correspond- find the dls " 

1 J ° ^ x tance of any 

ing to any other planet. Thus, for the distance of Venus, planet from 
we multiply 94880706 by .72333 ; and the result is the suu in 

miles 

68629960 miles, for the distance of Venus: and proceed, in 
the same manner, for the distance of any other planet. 

(129.) By observations taken on the transit of Venus, in To find the 
1769, it was concluded that the horizontal parallax of that y enus 
planet was 30". 4; and its semidiameter, at the same time, 
was 29".2. Hence (Art. 127), 304 : 292 : : 7912 : to a 
fourth term; which gives 7599 miles for the diameter of 
Venus. 

(130.) We cannot observe the horizontal parallax of Ju- Parallax 
piter, Saturn, or any other very remote planet: if known at cannotbeob-* 
all, it becomes known by computation; but the parallax can served, 
be known, when the real distance is known ; and, by Kepler's 
third law, and the solar parallax, we do know all the planetary 
distances ; and can, of course, compute any particular hori- 
zontal parallax. 

For the horizontal parallax of Jupiter, when at a distance 
from the earth equal to its mean distance from the sun, we 
proceed as follows : 

The parallax, or the semidiameter of the earth, when seen 
at the distance of the sun, is 8". 6. When seen from a greater 
distance, the angle would be proportionally less. 

Put h equal to the horizontal parallax of Jupiter ; then we 

have, - 5.202776 : 1 : : 8".6 : h; or h=~^-~. 

5.202776 

From this, we perceive, that if we divide the sun's horizontal How to 
parallax by the ratio of a planets distance from the sun, the com P ute the 
quotient will be the horizontal parallax of the planet, when at a the planet. 
distance from the earth equal to its mean distance from the sun. 

(131.) To find the diameter of a planet, in relation to the How to 
diameter of the earth, we have a similar proportion as in Art. Snd the real 

tnn -i .o 1 -i t i diameters of 

lib : and to find the diameter of Jupiter, we proceed as the planets. 
follows : 

The greatest apparent diameter of Jupiter, as seen from 
9 



130 ASTRONOMY. 

Chap. x. the earth, is 44".5; the least is 30". 1; therefore the mean, 
as seen from the sun, cannot be far from 37 ".3. and the semi- 
diameter 18".65; La Place says it is 18".35; and this value 
we shall use. Now, as in Art. 126, let d=7912, Z>= the 

8".6 
unknown diameter of Jupiter ; 90977ft * s ^ s nor i zon tal 

parallax, and 18".35 its corresponding semidiameter ; then, as 
in Art. 126, - 7912. : D : : -g-^g : 18.35; 

. Tnerefore jD= 7glg_Xl8.35x5.202776 = 7912><lm = 

8.0 
87900 miles. 

In the same manner, we may find the diameter of any 

other planet. 

Jupiter not ^ e have just seen that the diameter of Jupiter is 11.11 

spherical, times the diameter of the earth ; but this is the equatorial 

diameter of the planet. Its polar diameter is less, in the 

proportion of 167 to 177, as determined by the mean of many 

micrometrical measurements ; which proportion gives 82930 

miles, for the polar diameter of Jupiter. These extremes 

give the mean diameter of Jupiter, to the mean diameter of 

the earth, as 10.8 to 1. 

How to find (132.) But the magnitudes of similar bodies are to one 

the magm- ano ^ er as ^h e cu k es f their like dimensions ; therefore the 

tude of the 

planets. magnitude of Jupiter is to that of the earth, as (10.8) 3 to 
1, and from thence we learn that Jupiter is 1260 times 
greater than the earth. 

In the same manner we may find the magnitude of any 
other planet, and it is thus that their magnitudes have often 
been determined, and the results may be seen in a concise 
form, in Table IY, which gives a summary view of the solar 
system. 

The masses and attractions of the different planets will be 
investigated in physical astronomy, after we become acquain- 
ted with the theory of universal gravity. 



SOLAR SYSTEM. 131 



CHAPTER XI. 

A GENERAL DESCRIPTION OP THE PLANETS. 

( 133.) We conclude this section of astronomy by a brief chap. xi. 
description of the solar system, which we have purposely 
delayed lest we might interrupt the course of reasoning 
respecting the planetary motions. The reader is referred to 
Table IV, for a concise and comparative view of all the facts 
that can be numerically expressed ; and aside from these facts, 
little can be said by way of explanation or description. 

The fact, that the sun or a planet revolves on an axis, Facts reveal- 
must be determined by observing the motion of spots on the e .J spot 

•> o r on the sun or 

visible disc ; and if no spots are visible, the fact of revolution planets. 
cannot be ascertained.* But when spots are visible, their 
motion and apparent paths will not only point out the time 
of revolution, but the position of the axis. 

THE SUN. 

(134.) The sun is the central body in the system, of im- The snn the 
mense magnitude, comparatively stationary, the dispenser of Te P° sitor y of 
light and heat, and apparently the repository of that force 
which governs the motion of all other bodies in the system. 

" Spots on the sun seem first to have heen observed in the year 1611, 
since which time they have constantly attracted attention, and have 
been the subject of investigation among astronomers. These spots 
change their appearance as the sun revolves on its axis, and become 
greater or less, to an observer on the earth, as they are turned to, or 
from him; they also change in respect to real magnitude and number; 
one spot, seen by Dr. Herschel, was estimated to be more than six 
times the size of our earth, being 50000 miles in diameter. Some- 
times forty or fifty spots may be seen at the same time, and sometimes 
only one. They are often so large as to be seen with the naked eye ; 
this was the case in 1816. 

" In two instances, these spots have been seen to burst into several 
parts, and the parts to fly in several directions, like a piece of ice 
thrown upon the ground. 

* Mercury is an exception to this principle. 



132 ASTRONOMY. 

Chap. XI. " In respect to the nature and design of these spots, almost every 
astronomer has formed a different theory. Some have supposed them 
to be solid opaque masses of scoria?, floating in the liquid fire of the 
sun ; others as satellites, revolving round him, and hiding his light 
from us ; others as immense masses, which have fallen on his disc, and 
which are dark colored, because they have not yet become sufficiently 
heated. 

" Dr. Herschel, from many observations with his great telescope, 
concludes, that the shining matter of the sun consists of a mass of 
phosphoric clouds, and that the spots on his surface are owing to dis- 
turbances in the equilibrium of this luminous matter, by which open- 
ings are made through it. There are, however, objections to this 
theory, as indeed there are to all the others, and at present it can only be 
said, that no satisfactory explanation of the cause of these spots has 
been given." 

singular ( 135.) Mercury. This planet is the nearest to the sun, 
means of dis- aQ< j ^ ag ^ Qen ^ e SUD iect of considerable remark in the pre- 

covermg ro- * L 

tation. ceding pages. It is rarely visible, owing to its small size and 

proximity to the sun, and it never appears larger to the na- 
ked eye than a star of the fifth magnitude. 

Mercury is too near the sun to admit of any observations 
on the spots on its surface ; but its period of rotation has 
been determined by the variations in its horns — the same 
ragged corner comes round at regular intervals of time — 
24h. 5m. 

Times when The best time to see Mercury, in the evening, is in the 

Mercury may spring of the year, when the planet is at its greatest elonga- 
tion east of the sun. It will then be visible to the naked, 
eye about fifteen minutes, and will set about an hour and 
fifty minutes after the sun. When the planet is west of the 
sun, and at its greatest distance, it may be seen in the morn- 
ing, most advantageously in August and September. The 
symbol for the greatest elongation of Mercury, as written in 
the common almanacs, is y Gr. Elon. 
High moun- ( 136.) Venus. This planet is second in order from the sun, 

tams on Ve- an( j ^ n re l a tion to its position and motion, has been sufficiently 
described. The period of its rotation on its axis is 23h. 21m. 
The position of the axis is always the same, and is not at 
right angles to the plane of its orbit, which gives it a change of 
seasons. The tangent position of the sun's light across this 



SOLAR SYSTEM. 



IBS 




Chap. XI. 

Telescopic 
views of Ye- 



planet shows a very rough sur- 
face; indeed, high mountains. 
By the radiating and glimmer- 
ing nature of the light of this 
planet, we infer that it must 
have a deep and dense atmos- 
phere. 

( 137.) The Earth is the next planet in the system; but it The earth 
would be only formality to give any description of it in this a P lanet - 
place. As a planet, it seems to be highly favored above its 
neighboring planets, by being furnished with an attendant, The earth's 
the moon ; and insignificant as this latter body is, compared a 
to the whole solar system, it is the most important and in- 
teresting to the inhabitants of our earth. The two bodies, 
the earth and the moon, as seen from the sun, are very small : 
the former subtending an angle of about 17" in diameter, 
the latter about 4", and their distance asunder never greater 
than between seven and eight minutes of a degree. 

Contrary to the general impression, the moon's motion in 
absolute space is always concave toward the sun.* 

(138.) Mars — the first superior planet — is of a red color, Mars; his 
and very variable in its apparent magnitude. About every pe ^nce &c" 



* This may be shown thus — the moon is inside the earth's 
orbit from the last quarter to the first quarter, on an average 
14 days and 18 hours. During this time the earth moves in 
its orbit 14° 30'. Let ^g- 28 - 

L n F 'be a portion of the 
earth's orbit equal to 14° 30', L 
L the position of the earth at the First Quarter of the moon, 
and F its position at the Last Quarter. Draw the chord L F, 
and compute m n the versed sine of the arc 7° 15'. 

The mean radius of the earth's orbit is 397 times the ra- 
dius of the lunar orbit. A radius of 397 and an angle 7° 15' 
gives a versed sine of 3.49; but on this scale the distance 
from the earth to the moon is unity, or less than one third of 
nm; hence, the moon's path must be between the chord LF 
and the arc L n F — that is, always concave toward the sun. 

L 



The moon's 
motion con- 
cave toward 
the sun. 




134 ASTRONOMY. 

Chap. xi. other year, when it comes to the meridian, near midnight, it 

is then most conspicuous ; and the next year it is scarcely 

noticed by the common observer. 

■ "The physical appearance of 
Telescopic View of Mars. tvt • u , i_ ui tr- 
Mars is somewhat remarkable. His 

polar regions, when seen through 
a telescope, have a brilliancy so 
much greater than the rest of his 
disc, that there can be little doubt 
that, as with the earth so with 
this planet, accumulations of ice 
or snow take place during the win- 
ters of those regions. In 1781 
the south polar spot was extremely 
bright ; for a year it had not been 
exposed to the solar rays. The 
color of the planet most probably 
arises from a dense atmosphere which surrounds him, of the existence of 
which there is other proof depending on the appearance of stars as 
they approach him ; they grow dim and are sometimes wholly extin- 
guished as their rays pass through that medium." 

Apparentim- (139.) The next planet, as known to ancient astronomers? 
perfection in is Jupiter ; but its distance is so great beyond the orbit of 
system. jyj ars ^ ^^ ^ e Y0 ^ S p ace between the two had often been 
considered as an imperfection, and it was a general impression 
among astronomers that a planet ought to occupy that vacant 
space. 
Bode'siaw. Professor Bode, of Berlin, on comparing the relative dis- 
tances of the planets from the sun, discovered the following re- 
markable fact — that if we take the following series of numbers : 

0, 3, 6, 12, 24, 48, 96, 192, &c, 
and then add the number 4 to each, and we have, 
4, 7, 10, 16, 28, 52, 100, 196, &c, 
The reason and this last series of numbers very nearly, though not ex- 
no/be called ttC %> corresponds to the relative distances of the planets from 
a law. the sun, with the exception of the number 28. This is 
sometimes called Bode's law ; but remarkable as it certainly 
is, it should not be dignified by the term law, until some bet- 
ter account of it can be given than its mere existence ; for, 
at present, all that can be said of it is, " here is an astonishing 



SOLAR SYSTEM. 



135 



coincidence." But, mere accident as it may be, it suggested chap. xi. 
the possibility of some small, undiscovered planet revolving A bold h 
in this region, and we can easily imagine the astonishment of pothesis. 
astronomers, on finding four in place of one, revolving in 
orbits tolerably well corresponding to this law, or rather co- 
incidence. Had they even found but one, it would seem to 
indicate something more than mere coincidence ; but finding 
four, proves the series to be simply accidental — unless the 
four or more planets there discovered were originally one 
planet ; and then came the inquiry, is not this the case ? Thus 
originated the idea that these new and newly discovered small 
planets are but fragments of a larger one, which formerly cir- 
culated in that interval, and was blown to pieces by some 
internal explosion — and we shall examine this hypothesis in a 
text note, under physical astronomy. 

The names of these planets, in the order of the times of their 
discovery, are, Ceres, Pallas, Juno, Vesta. The order of their 
distances from the sun, is Vesta, Juno, Ceres, Pallas. 



Planets. 


Names of Dis- 
coverers. 


Residence of Discoverers. 


Date of Discovery. 


Ceres . . . 

Pallas... 

Juno . . . 

i Vesta . . . 


M. Piazzi, 
Dr. Olbers, 
M. Harding, 
Dr. Olbers, 


Palermo, Sicily, 
Bremen, Germany, 
Lilienthal, near Bremen, 
Bremen, 


1st Jan., 1801. 
28th Mar., 1802. 

1st Sept. 1804. 
29th Mar., 1807. 



If a planet has really burst, it is but reasonable to suppose 
that it separated into many fragments ; and, agreeably to this 
view of the subject, astronomers have been constantly on the 
alert for new planets, in the same regions of space ; and every R ece nt 
discovery of the kind greatly increases the probability of the discoveries 
theory. The following very recent discoveries are said to have ™°™ ^ the ° 
been made, but the elements of the orbits are not regarded as sis. 
sufficiently accurate to demand a place in the table. 

On the 8th of December, 1845, Mr. Hencke, of Dreisen, 
claims to have discovered a planet which he calls Astrea; 
and the same observer also claims another, discovered in 
1847, called Hebe. His success induced others to a like exa- ets discover- 
mination, and a Mr. Hind, of London, within the past year, and 1846# 
10 



New plan- 



186 



ASTRONOMY. 



chap. xi. 1848, claims a seventh and eighth asteroid, named Iris and 
Flora. 

Thus we have eight miniature worlds, supposed to have 
once composed a planet ; and if the four last named are veri- 
table discoveries, we shall soon have the elements of their 
orbits in an unquestionable shape. 

The elements of the orbits of the four known asteroids, as 
given for the epoch 1820, are not as accurate as the follow- 
ing, which were deduced from the Nautical Almanac for 1846 
and 1847 ; which have been corrected from more modern, 
extended, and accurate observations. (Epoch Jan., 1847.) 

On account of the small magnitude of these new planets, 
and their recent discovery, nothing is known of them save 
the following tabular facts, and these are only approximation 
to the truth. 



Planets. 


Sidereal 
Revolutions. 


Mean Distance from 
the Sun. 


Eccentricity of 
Orbits. 


1 Pallas 


Days. 
1324. 289 
1594. 721 
1683. 064 
1685. 162 


2. 36120 
2. 66514 
2. 76910 
2. 77125 


0. 08913 
0. 25385 
0. 07844 
0. 24050 


Planets. 


Longitude of 
Ascending Node. 


Inclination of 
Orbits. 


Longitude of 
Perihelion. 


Vesta 

Pallas 


O ' " 

103 20 47 

170 53 

80 47 56 

172 42 38 


O ' " 

7 8 29 
13 2 53 
10 37 17 
34 37 42 


O ' " 

251 4 34 

54 18 32 

147 25 41 

121 20 13 



Object of 
Fig. 29. 



( 140.) With the two elements, the longitude of the ascend- 
ing nodes, and the inclination of the orbits to the ecliptic, we 
are enabled to give a general projection of these orbits around 
the celestial sphere, in relation to the ecliptic, as represented 
on page 37 : and our object is to show that there are two 
points in the heavens, nearly opposite to each other, near to 
which all these planets pass. One of these points is about 
the longitude of 185 degrees, and the latitude of 15 degrees 
north ; and the other is the opposite point on the celestial 
sphere. If these planets are but fragments of one original 
planet, which burst or exploded by its internal fixes, from that 



SOLAR SYSTEM 



137 



moment they must have 
started from the same 
point, and the orbits of all 
have one common distance 
from the sun ; and for 
ages after such a catas- 
trophe, these fragments 
must have had nearly a 
common node; and the 
fact that they do not, at 
'present, pass through a 
common point, nor have 
a common node, does not 
prove that they were not 
originally in one body; 
for, owing to mutual dis- 
turbances, and the dis- 
turbances of other pla- 
nets, the nodes must 
change positions; and the 
longer axis of the orbits, 
especially the very ec- 
centric ones, must change 
positions; and now (after 
we know not how many 
ages), it is not incon- 
sistent with the theory 
of an explosion, that we 
find the orbits as they 
are. 

The hypothesis that 
these planets were ori- 
ginally one, and must, 
therefore, have two com- 
mon points in the hea- 
vens near which they 
must all pass, led to the 
discovery of Juno and 




Chap. XI. 

Where the 
original pla- 
net must 
have explod- 
ed, if the 
hypothesis 
of an original 
planet is true 



Fisr. '-2lL 



138 ASTRONOMY. 

Chap. xi. Vesta, by carefully observing these two portions of the 
heavens. 

The apparent diameters of these planets are too small to 
be accurately measured; and therefore we have only a very 
rough or conjectural knowledge of their real diameters. 

All of these planets are invisible to the naked eye, except 
Vesta, which sometimes can be seen as a star of the 5th or 
6th magnitude. 

(141.) Jupiter. We now come to the most magnificent 
planet in the system — the well-known Jupiter — which is 
nearly 1300 times the magnitude of the earth. 
Jupiter's The disc of Jupiter is always observed to be crossed, in an 
eastern and western direction, by dark bands, as represented 
in Fig. 30. 

Fig. 30. — Telescopic View of Jupiter. 



belts 




" These belts are, however, by no means alike at all times ; they 
vary in breadth and in situation on the disc (though never in their 
general direction). They have even been seen broken up, and distri- 
buted over the whole face of the planet : but this phenomenon is ex- 
tremely rare. Branches running out from them, and subdivisions, as 
represented in the figure, as well as evident dark spots, like strings of 
clouds, are by no means uncommon ; and from these, attentively 
watched, it is concluded that this planet revolves in the surprisingly 
Diurnal re- short period of 9 h. 55 m. 50 s. (sid. time), on an axis perpendicular to 
the direction of the belts. Now, it is very remarkable, and forms a 
most satisfactory comment on the reasoning by which the spheroidal 
figure of the earth has been deduced from its diurnal rotation, that the 
outline of Jupiter's disc is evidently not circular, but elliptic, being 
considerably flattened in the direction of its axis of rotation. 



volution 



SOLAR SYSTEM. 139 

" The parallelism of the belts to the equator of Jupiter, their occa- chap, XI, 

sional variations, and the appearances of spots seen upon them, render ~~ 

it extremely probable that they subsist in the atmosphere of the planet, of JuDiter 
forming tracts of comparatively clear sky, determined by currents ana- 
logous to our tradewinds, but of a much more steady and decided cha- 
racter, as might indeed be expected from the immense velocity of its 
rotation. That it is the comparatively darker body of the planet which 
appears in the belts, is evident from this, — that they do not come up 
in all their strength to the edge of the disc, but fade away gradually be- 
fore they reach it. 

(142.) "When Jupiter is viewed with a telescope, even of moderate Jupiter's 
power, it is seen accompanied by four small stars, nearly in a straight satellites, 
line parallel to the ecliptic. These always accompany the planet, and 
are called its Satellites. They are continually changing their positions 
with respect to one another, and to the planet, being sometimes all to 
the right, and sometimes all to the left ; but more frequently some on 
each side. The greatest distances to which they recede from the planet, 
on each side, are different for the different satellites, and they are thus 
distinguished : that being called the First satellite, which recedes to the 
least distance ; that the Second, which recedes to the next greater dis- 
tance, and so on. The satellites of Jupiter were discovered by Galileo, 
in 1610. 

" Sometimes a satellite is observed to pass between the sun and Ju- 
piter, and to cast a shadow which describes a chord across the disc. 
This produces an eclipse of the sun, to Jupiter, analogous to those 
which the moon produces on the earth. It follows that Jupiter and 
its satellites are opake bodies, which shine by reflecting the sun's 
light. 

" Careful and repeated observations show that the motions of the satel- 
lites are from west to east, in orbits nearly circular, and making small 
angles with the plane of Jupiter's orbit. Observations on the eclipses 
of the satellites make known their synodic revolutions, from which 
their sidereal revolutions are easily deduced. From measurements of 
the greatest apparent distances of the satellites from the planet, their 
real distances are determined. 

" A comparison of the mean distances of the satellites, with their side- 
real revolutions, proves that Kepler's third law, with respect to the 
planets, applies also to the satellites of Jupiter. The squares of their 
sidereal revolutions are as the cubes of their mean distances from the 
planet. 

" The planets Saturn and Uranus are also attended by satellites, and 
the same law has place with them." 

( 143.) By the eclipses of Jupiter's satellites, the progres- nat r u ° r = r essl ™ 
sive nature of light was discovered ; which we illustrate in light, 
the following manner : 



140 ASTRONOMY. 

Chap. XI. Fig. 31. r // ?»»»»— ». 




Let S (Fig-. 31) represent the sun, J Jupiter, dearth, and m Jupiter's 
first satellite. By careful and accurate observations astronomers have 
decided that the mean revolution of this satellite round its primary, is 
performed in 42 h. 28 m. and 35 s. ; that is, the mean time from one 
eclipse to another. 
Velocity of But when the earth is at E, and moving in a direction toward, or 
light, how nearly toward, the planet as represented in the figure, the mean time 
etermmed. De t ween two consecutive eclipses is shortened about 15 seconds ; and 
we can explain this on no other hypothesis than that the earth has ad- 
vanced and met the successive progression of light. When the earth 
is in position as respects the sun and Jupiter, as represented in our 
figure at E', and moving from Jupiter, then the interval between two 
consecutive eclipses of Jupiter's first satellite is prolonged or increased 
about 15 seconds. 

But during the interval of one revolution of Jupiter's first satellite, 
the earth moves in its orbit about 2880000 miles ; this, divided by 15, 
gives 192000 miles for the motion of light in one second of time ; and 
this velocity will carry light from the sun to the earth in about eight 
and one-fourth minutes. 

Longitude (144.) As an eclipse of one of Jupiter's satellites maybe 

found by the geen f rom a ]} p] aces where the planet is there visible, two 

Jupiter's sa- observers viewing it will have a signal for the same moment, 

teiiites. at their respective places ; and their difference in local time 

will give their difference in longitude. For example, if one 

observer saw one of these eclipses at 10 h.in the evening, and 

another at 8 h. 30 m., the difference of longitude between the 

observers would be 1 h. 30 m. in time, or 22° 30' of arc. 

The absolute time that the eclipse takes place, is the same 
to all observers ; and he who has the latest local time is the 
most eastward. 

These eclipses cannot be observed at sea, by reason of the 
motion of the vessel. 



SOLAR SYSTEM. 141 

(145.) Saturn. The next planet in order of remoteness Chap. xi. 

from the sun, is Saturn, the most wonderful object in the Satum- 

solar system. Though less than Jupiter, it is about 79000 hls nngs ' 

miles in diameter, and 1000 times greater than our earth. 

" This stupendous globe, besides being attended by no less than seven 
satellites, or moons, is surrounded with two broad, flat, extremely thin 
rings, concentric with the planet and with each other ; both lying in 
one plane, and separated by a very narrow interval from each other 
throughout their whole circumference, as they are from the planet by 
a much wider. The dimensions of this extraordinary appendage are 
as follows : 

Exterior diameter of exterior ring, = 176418. 

Interior ditto, = 155272. 

Exterior diameter of interior ring, = 151690. 

Interior ditto, = 117339. 

Equatorial diameter of the body, = 79160. 

Interval between the planet and interior ring, = 19090. 

Interval of the rings = 1791. 

Thickness of the rings not exceeding, = 100. Dimensions 

Fig. 32. — Telescopic View of Saturn. 




" The figure represents Saturn surrounded by its rings, and having its The rings 
body striped with dark belts, somewhat similar, but broader and less are °P ake ' 
strongly marked than those of Jupiter, and owing, doubtless, to a simi- 
lar cause. That the ring is a solid opake substance, is shown by its 
throwing its shadow on the body of the planet, on the side nearest the 
sun, and on the other side receiving that of the body, as shown in the 
figure. From the parallelism of the belts with the plane of the ring, 
it may be conjectured that the axis of rotation of the planet is perpen- 
dicular to that plane ; and this conjecture is confirmed by the occa- 
sional appearance of extensive dusky spots on its surface, which when 
watched, like the spots on Mars or Jupiter, indicate a rotation in 10 h. 
09 m. 17 s. about an axis so situated. 

" It will naturally be asked how so stupendous an arch, if composed 
of solid and ponderous materials, can be sustained without collapsing 



142 ASTRONOMY. 

Chap. XI. and tailing in upon the planet ? The answer to this is to be found in 
ZZ ~~ . . a swift rotation of the ring in its own plane, which observation has 
lity of the detected, owing to some portions of the ring being a little less bright 
rings. than others, and assigned its period at 10 h. 29 m. 17 s., which, from 

what we know of its dimensions, and of the force of gravity in the 
Saturnian system, is very nearly the periodic time of a satellite revolv- 
ing at the same distance as the middle of its breadth. It is the centri- 
fugal force, then, arising from this rotation, which sustains it ; and, 
although no observation nice enough to exhibit a difference of periods 
between the outer and inner rings have hitherto been made, it is more 
than probable that such a difference does subsist as to place each inde- 
pendently of the other in a similar state of equilibrium. 
The rings " Although the rings are, as we have said, very nearly concentric 
revolve a- w ith the body of Saturn, yet recent micrometrical measurements, of 

round the ex t reme delicacy, have demonstrated that the coincidence is not mathe- 
planet like . ■ „ , , . „ . „ , .„ . 

... matically exact, but that the center of gravity ot the rings oscillates 

round that of the body, describing a very minute orbit, probably under 

laws of much complexity. Trifling as this remark may appear, it is 

of the utmost importance to the stability of the system of the rings. 

Supposing them mathematically perfect in their circular form, and 

exactly concentric with the planet, it is demonstrable that they would 

form (in spite of their centrifugal force) a system in a state of unstable 

equilibrium, which the slightest external power would subvert — not by 

causing a rupture in the substance of the rings — but by precipitating 

them, unbroken, on the surface of the planet. For the attraction of 

such a ring or rings on a point or sphere eccentrically situate within 

them, is not the same in all directions, but tends to draw the point or 

sphere toward the nearest part of the ring, or away from the center. 

Hence, supposing the body to become, from any cause, ever so little 

eccentric to the ring, the tendency of their mutual gravity is, not to 

correct, but to increase this eccentricity, and to bring the nearest parts 

of them together." 

Uranus alias (146.) Uranus. The next planet, beyond Saturn, was 
Herschei. discovered by Sir W. F. Herschel, in 1781, and, for a time, 
was called Herschel, in honor of its discoverer ; but, accord- 
ing to custom, the name of a heathen deity has been substi- 
tuted, and the planet is now called Uranus — the father of 
Saturn. 
This l et ^^ s planet is rarely to be seen, without a telescope. In a 
rarely visible clear night, and in the absence of the moon, when in a favor- 
to the naked ^q position above the horizon, it may be seen as a star of 
about the 6th magnitude. Its real diameter is about 35000 
miles, and about 80 times the magnitude of the earth. 



SOLAR SYSTEM. . 143 

The existence of this planet was suggested by some Chap. xi. 
of the perturbations of Saturn ; which could not be accounted 
for by the action of the then known planets ; but it does not 
appear that any computations were made, as a guide to the 
place where the unknown disturbing body ought to exist ; and, 
as far as we know, the discovery by Herschel was mere 
accident. 

But not so with the planet Neptune, discovered in the Facts led 
latter part of September, 1846, by a French astronomer, Le- t0 the disc0 ' 
verrier ; and also a Mr. Adams, of Cambridge, England, who has tune# 
put in his claim as the discoverer. The truth is, that the 
attention of the astronomers of Europe had been called to 
some extraordinary perturbations of Uranus ; which could not 
be accounted for without supposing an attracting body to be 
situated in space, beyond the orbit of Uranus ; and so distinct 
and clear were these irregularities, that both geometers, Le- 
verrier and Adams, fixed on the same region of the heavens, 
for the then position of their hypothetical planet ; and by dili- 
gent search, the planet was actually discovered about the 
same time, in both France and England. 

At present, we can know very little of this planet ; and 
according to the best authority I can gather, its longi- 
tude, January 1, 1847, was 327° 24'. Mean distance from 
the sun, 30.2 ( the earth's distance being unity) ; period of 
revolution 166 years. Eccentricity of orbit 0.0084; mass, 

1 
23000"' 

According to Bode's law, the distance of the next planet 
from the sun, beyond Uranus, must be 38.8 ; and if Neptune 
really is at 30.2, it shows Bode's law to be only a remarkable 
coincidence ; for there can be no exceptions to positive physi- 
cal laws. 

" We shall close this chapter with an illustration calculated to convey How ;o 
to the minds of our readers a general impression of the relative magni- btain a cor- 
tudes and distances of the parts of our system. Choose any well- rect conce p- 
leveled field or bowling green. On it place a globe, two feet in diame- tion of the 
ter ; this will represent the sun ; Mercury will be represented by a grain solar system 
of mustard seed, on the circumference of a circle 164 feet in diameter, 
for its orbit ; Venus a pea, on a circle 284 feet in diameter ; the earth 



144 ASTRONOMY. 

Chap. XI. also a pea. on a circle of 430 feet ; Mars a rather large pin's head, on a 
circle of 654 feet; Juno, Ceres, Vesta, and Pallas, grains of sand, in 
orbits of from 1000 to 1200 feet ; Jupiter a moderate-sized orange, in a 
circle nearly half a mile across; Saturn a small orange, on a circle of 
four-fifths of a mile ; and Uranus a full-sized cherry, or small plum, 
upon the circumference of a circle more than a mile and a half in dia- 
meter. As to getting correct notions on this subject by drawing circles 
on paper, or, still worse, from those very childish toys called orreries, 
it is out of the question. To imitate the motions of the planets in the 
View of above-mentioned orbits, Mercury must describe its own diameter in 41 

the planetary seconds ; Venus, in 4 m. 14 s. ; the earth, in 7 minutes ; Mars, in 4 m. 

motions. 48 s . . Jupiter, in2h. 56 m. ; Saturn, in3h. 13m. ; and Uranus, in 2h. 
16 m." — HerscheVs Astronomy. 



CHAPTER XII. 

ON COMETS. 

chap. xii. (147.) Besides the planets, and their satellites, there are 
Comets great numbers of other bodies, which gradually come into 

formerly in- v i eWj increasing in brightness and velocity, until they attain 

ror a maximum, and then as gradually diminish, pass off, and are 

lost in the distance. 
Knowledge " These bodies are comets. From their singular and unusual appear- 

banishes ance, they were for a long time objects of terror to mankind, and were 

dread. regarded as harbingers of some great calamity. 

" The luminous train which accompanied them was particularly 
alarming, and the more so in proportion to its length. It is but little 
more than half a century since these superstitious fears were dissipated 
by a sound philosophy ; and comets, being now better understood, 
excite only the curiosity of astronomers and of mankind in general. 
These discoveries which give fortitude to the human mind are not 
among the least useful. 

" It was formerly doubted whether comets belonged to the class of 
heavenly bodies, or were only meteors engendered fortuitously in the 
air by the inflammation of certain vapors. Before the invention of the 
telescope, there were no means of observing the progressive increase 
and diminution of their light. They were seen but for a short time, 
and their appearance and disappearance took place suddenly. Their 
light and vapory tails, through which the stars were visible, and their 
whiteness often intense, seemed to give them a strong resemblance to 
those transient fires, which we call shooting stars. Apparently, they 
differed from these only in duration. They might be only composed 



COMETS. 145 

of a more compact substance capable of retarding for a longer time Chap. XII. 
their dissolution. But these opinions are no longer maintained ; more 
accurate observations have led to a different theory. 

"All the comets hitherto observed have a small parallax,* which places Parallax of 
them far beyond the orbit of the moon ; they are not, therefore, formed comets, 
in our atmosphere. Moreover, their apparent motion among the stars 
is subject to regular laws, which enable us to predict their whole course 
from a small number of observations. This regularity and constancy 
evidently indicate durable bodies ; and it is natural to conclude that 
comets are as permanent as the planets, but subject to a different kind 
of movement. 

" When we observe these bodies with a telescope, they resemble a mass Comets are 

of vapor, at the center of which is commonly seen a nucleus more or a PP arentlv 

less distinctly terminated. Some, however, have appeared to consist „ 

* of vapor, 

of merely a light vapor, without a sensible nucleus, since the stars are 

visible through it. During their revolution, they experience progres- 
sive variations in their brightness, which appear to depend upon their 
distance from the sun, either because the sun inflames them by its heat, 
or simply on account of a stronger illumination. When their bright- 
ness is greatest, we may conclude from this very circumstance that 
they are near their perihelion. Their light is at first very feeble, but 
becomes gradually more vivid, until it sometimes surpasses that of the 
brightest planets ; after which it declines by the same degrees until it 
becomes imperceptible. We are hence led to the conclusion that 
comets, coming from the remote regions of the heavens, approach, in 
many instances, much nearer the sun than the planets, and then recede 
to much greater distances. 

" Since comets are bodies which seem to belong to our planetary Orbits of 
system, it is natural to suppose that they move about the sun like comets « 
planets, but in orbits extremely elongated. These orbits must, there- 
fore, still be ellipses, having their foci at the center of the sun, but 
having their major axes almost infinite, especially with respect to us, 
who observe only a small portion of the orbit, namely, that in which 
the comet becomes visible as it approaches the sun. Accordingly the 
orbits of comets must take the form of a parabola, for we thus designate 
the curve into which the ellipse passes, when indefinitely elongated. 

" If we introduce this modification into the laws of Kepler, which 

* The parallaxes of comets are known to be small, by two observers, 
at distant stations on the earth, comparing their observations taken 
on the same comet at near the same time. At the times the observa- 
tions are made, neither observer can know how great the parallax is. 
It is only afterward, when comparisons are made, that judgment, in 
this particular, can be formed ; and it is not common that any more 
definite conclusion can be drawn, than that the parallax is small, and, 
of course, the body distant. 

10 M 



146 



ASTRONOMY. 



Chap. XII. 



Comets des- 
cribe equal 
areas in e- 
qual times. 



Three obser- 
vations suffi- 
cient to find 
the orbit of a 
comet. 



relate to the elliptical motion, we obtain those of the parabolic motion 
of comets. 

" Hence it follows that the areas described by the same comet, in its 
parabolic orbit, are proportional to the times. The areas described by 
different comets in the same time, are proportional to the square roots 
of their perihelion distances. 

" Lastly, if we suppose a planet moving in a circular orbit, whose 
radius is equal to the perihelion distance of a comet, the areas described 
by these two bodies in the same time, will be to each other as 1 to 

/2. Thus are the motions of comets and planets connected. 
" By means of these laws we can determine the area described by 
a comet in a given time after passing the perihelion, and fix its posi- 
tion in the parabola. It only remains then to bring this theory to the 
test of observation. Now we have a rigorous method of verifying it, 
by causing a parabola to pass through several observed places of a 
comet, and then ascertaining whether all the others are contained in it. 

" For this purpose three observations are requisite. If we observe 
the right ascension and declination of a comet at three different 
times, and thence deduce its geocentric longitude and latitude, we 
shall have the direction of three visual rays drawn at these times from 
the earth to the comet, and in the prolongation of which it must 
necessarily be found. The corresponding places of the sun are also 
known ; it remains then to construct a parabola, having its focus at 
the center of the sun, and cutting the visual rays in points, the inter- 
vals of which correspond to the number of days between the obser- 
vations. 

" Or if we suppose the earth in mo- 
tion and the sun at rest, let T, T', T", 
represent three successive positions of 
the earth, and TC, T'C, T"C", three 
visual rays drawn to the comet. The 
question is to find a parabola CC'C", 
having its focus in S at the center of 
the sun, and cutting the three visual 
rays conformably to the conditions re- 
quired. 

Th b'tofa " These conditions are more than sufficient to determine completely 
comet found the elements of the parabolic motion, that is, the perihelion distance 
by these ob- of the comet, the position of the perihelion, the instant of passing this 
servations. point, the inclination of the orbit to the ecliptic, and the position of 
its nodes. These five elements being known, we can assign the posi- 
tion of the comet for any time whatever, and compare it with the 
results of observation. But the calculation of the elements is very 
difficult, and can be performed only by a very delicate analysis, which 
cannot here be made known. 




COMETS. 147 

"About 120 comets have been calculated upon the theory of the Chap. xn. 

parabolic motion, and the observed places are found to answer to such 

a supposition. We can have no doubt, therefore, that this is conform- T ' 

111 /* ttt i .i Inclinations 

able to the law of nature. We have thus obtained precise knowledge f tne j r or _ 

of the motions of these bodies, and are enabled to follow them in space, bits. 

This discovery has given additional confirmation to the laws of Kepler, 

and led to several other important results. 

" Comets do not all move from west to east like the planets. Some 
have a direct, and some a retrograde motion. 

" Their orbits are not comprehended within a narrow zone of the 
heavens, like those of the principal planets. They vary through all 
degrees of inclination. There are some whose plane is nearly coinci- 
dent with that of the ecliptic, and others have their planes perpendicular 
to it. 

" It is farther to be observed that the tails of comets begin to appear, 
as the bodies approach near the sun ; their length increases with this 
proximity, and they do not acquire their greatest extent, until after 
passing the perihelion. The direction is generally opposite to the sun, 
forming a curve slightly concave, the sun on the concave side. 

" The portion of the comet nearest to the sun must move more rapidly 
than its remoter parts, and this will account for the lengthening of the 
tail. 

" The tail is, however, by no means an invariable appendage of Some com- 
comets. Many of the brightest have been observed to have short and ets have no 
feeble tails, and not a few have been entirely without them. Those ta ^ s « 
of 1585 and 1763 offered no vestige of a tail; and Cassini describes the 
comet of 1682 as being as round and as bright as Jupiter. On the other 
hand, instances are not wanting of comets furnished with many tails, 
or streams of diverging light. That of 1744 had no less than six, 
spread out like an immense fan, extending to a distance of nearly 30 
degrees in length. 

" The smaller comets, such as are visible only in telescopes, or with 
difficulty by the naked eye, and which are by far the most numerous, 
offer very frequently no appearance of a tail, and appear only as round 
©r somewhat oval vaporous masses, more dense toward the center; 
where, however, they appear to have no distinct nucleus, or anything 
which seems entitled to be considered as a solid body. 

" The tail of the comet of 1456 was 60 degrees long. That of 1618, others have 
100 degrees, so that its tail had not all risen when its head reached the several tails, 
middle of the heavens. The comet of 1680 was so great, that though 
its head set soon after the sun, its tail, 70 degrees long, continued visi- 
ble all night. The comet of 1689 had a tail 68 degrees long. That of 
1769 had a tail more than 90 degrees in length. That of 1811 had a 
tail 23 degrees long. The recent comet of 1843 had a tail 60 degrees 
in length." 

The following figure gives a telescopic view of the comet of 1811. 



148 



ASTRONOMY. 



Chap. XIL 

Elements 
of comets 
how deter- 
mined. 



"When we have determined the elements of a comet's orbit, we com- 
pare them with those of comets before observed, and see whether there 
is an agreement with respect to any of them. If there is a perfect 
identity as to the elements, we should have no hesitation in concluding 
that they belonged to different appearances of the same comet. But 
this condition is not rigorously necessary ; for the elements of the 
orbit may, like those of other heavenly bodies, have undergone changes 
from the perturbations of the planets or their mutual attractions. Con- 
sequently, we have only to see whether the actual elements are nearly 
the same with those of any comet before observed, and then, by the doc- 
trine of chances, we can judge what reliance is to be placed upon this 
resemblance." Comet of 18u> 




verified. 



Dr.Halley's "Dr. Halley remarked that the comets observed in 1531, 1607, 1682, 
prediction had nearly the same elements ; and he hence concluded that they be- 
longed to the same comet, which, in 151 years, made two revolutions, 
its period being about 76 years. It actually appeared in 1759, agreea- 
bly to the prediction of this great astronomer ; and again in 1832. by 
the computation of several eminent astronomers. According to Kep- 
ler's third law, if we take for unity half the major axis of tne earth's 
Particulars orbit, the mean distance of this comet must be equal to the cube root 
of comets. f the square of 76, that is, to 17.95. The major axis of its orbit must, 
therefore, be 35.9 ; and as its observed perihelion distance is found to 
be 0.58. it follows that its aphelion distance is equal to 35.32. It 



335° 


249° 


13° 


13° 


157° 


108° 


2.2 


3.6 


1.2 


2.4 


3.29 


6.74 


May 4. 


Nov. 27 


1832 


1832 



COMETS. 149 

departs, therefore, from the sun to thirty-five times the distance of the Chap. XII. 
earth, and afterward approaches nearly twice as near the sun as the 
earth is, thus describing an ellipse extremely elongated. 

"The intervals of its return to its perihelion are not constantly the 
same. That between 1531 and 1607 was three months longer than 
that between 1607 and 1682 ; and this last was 18 months shorter than 
the one between 1682 and 1759. It appears, therefore, that the motions 
of comets are subject to perturbations, like those of the planets, and to 
a much more sensible degree. 

" Elements of the Orbits of the three Comets, which have appeared ac- 
cording to prediction, taken from the work of Professor Littrow. 

Halley. Encke. Biela. 
Longitude of the ascending node, - 54° 

Inclination of the orbit to the ecliptic, 162° 
Longitude of the perihelion, - - 303° 
Greatest semidiameter, that of the earth ) , q 

being called 1, - - - - ) 
Least semidiameter.. - 4.6 

Time of revolution in years, - 76 

Nov. 16. 
Time of the perihelion passage, - 1835 

" The comets of Encke and Biela move according to the order of the 
signs of the zodiac, or have their motions direct; the motion of that 
of Halley is retrograde. 

"Comets, in passing among and near the planets, are materially Jupiter, 

drawn aside from their courses, and in some cases have their orbits en- andhissatel- 
tirely changed. This is remarkably the case with Jupiter, which seems, s ' a great 
by some strange fatality, to be constantly in their way, and to serve as !. b . 

a perpetual stumbling-block to them. In the case of the remarkable comets . 
comet of 1770, which was found by Lexell to revolve in a moderate 
ellipse in the period of about five years, and whose return was pre- 
dicted by him accordingly, the prediction was disappointed by the comet 
actually getting entangled among the satellites of Jupiter, and being 
completely thrown out of its orbit by the attraction of that planet, and 
forced into a much larger ellipse. By this extraordinary renconter, 
the motions of the satellites suffered not the least perceptible derangement — 
a sufficient proof of the smallness of the comet's mass." 

The comet of 1456, represented as having a tail of 60° in length, is 
now found to be Halley's comet, which has made several returns — 
in 1531, 1607, 1682, 1759, and recently, in 1835. In 1607 the tail was 
said to have been over 30° in length ; but in 1835 the tail did not ex- 
ceed 12° Does it lose substance, or does the matter composing tho 
tail condense ? or, have we received only exaggerated and distorted 
accounts from the earlier times, such as fear, superstition, and awe, 
always put forth ? We ask these questions, but cannot answer them. 



150 ASTRONOMY. 

Chap. XII. The following cut represents the appearance of the comet of 
1819. 




Fears en- " Professor Kendall, in his Uranography, speaking of the fears occa- 
tertained, by gjoned by comets, says: "Another source of apprehension, with regard 
some, t at ^ Q come t S} ar i ses from the possibility of their striking our earth. It is 
quite probable that even in the historical period the earth has been 
come into enveloped in the tail of a comet. It is not likely that the effect would 
collision with be sensible at the time. The actual shock of the head of a comet against 
our earth. the earth is extremely improbable. It is not likely to happen once in 
a million of years. 

" If such a shock should occur, the consequences might perhaps be 
very trivial. It is quite possible that many of the comets are not 
heavier than a single mountain on the surface of the earth. It is well 
known that the size of mountains on the earth is illustrated by com- 
paring them to particles of dust on a common globe." 



CHAPTER XIII. 

ON THE PECULIARITIES OF THE FIXED STARS. 

Cha p, xii i. j? on ^ e f ac |; S as contained in the subject matter of this 
chapter, we must depend wholly on authority ; for that reason 
we give only a compilation, made in as brief a manner as the 
nature of the subject will admit. 

In the first part of this work it was soon discovered that 
the fixed stars were more remote than the sun or planets ; 
and now, having determined their distances, we may make 
further inquiries as to the distances to the stars, which will 



FIXED STARS. 151 

give some index by which to judge of their magnitudes, nature, chap. xin. 
and peculiarities. 

" It would be idle to inquire whether the fixed stars have a sensible Base from 
parallax, when observed from different parts of the earth. We have wllich to 
already had abundant evidence that their distance is almost infinite. It measure to 

the st3.rs 

is only by taking the longest base accessible to us, that we can hope to 
arrive at any satisfactory result. 

"Accordingly, we employ the major axis of the earth's orbit, which is 
nearly 200 millions of miles in extent. By observing a star from the 
two extremities of this axis, at intervals of six months, and applying a 
correction for all the small inequalities, the effect of which we have 
calculated, we shall know whether the longitude and latitude are the 
same or not at these two epochs. 

" It is obvious, indeed, that the star must appear more elevated above Annual 

the plane of the ecliptic when the earth is in the part of its orbit which parallax. 
is nearest to the star, and more depressed when the contrary takes 
place. The visual rays drawn from the earth to the star, in these two 
positions, differ from the straight line drawn from the star to the center 
of the earth's orbit ; and t^e angle which either of them forms with 
this straight line, is called the annual parallax. 

" As the earth does not pass suddenly from one point of its orbit to The effect 
the opposite, but proceeds gradually, if we observe the positions of a of a sensible 
star at the intermediate epochs, we ought, if the annual parallax is sen- para ' 
sible, to see its effects developed in the same gradual manner. For 
example, if the star is placed at the pole of the ecliptic, the visual rays 
drawn from it to the earth, will form a conical surface, having its apex 
at the star, and for its base, the earth's orbit. This conical surface 
being produced beyond the star, will form another opposite to the first, 
and the intersection of this last with the celestial sphere, will constitute 
a small ellipse, in which the star will always appear diametrically oppo- 
site to the earth, and in the prolongation of the visual rays drawn to 
the apex of the cones. 

" But notwithstanding all the pains that have been taken to multiply The annual 
observations, and all the care that has been used to render them per- parallaxmust 
fectly exact, we have been able to discover nothing which indicates, be Iess than 
with certainty, even the existence of an annual parallax, to say nothing 
of its magnitude. Yet the precision of modern observations is such, 
that if this parallax were only 1", it is altogether probable that it would 
not have escaped the multiplied efforts of observers, and especially those 
of Dr. Bradley, who made many observations to discover it, and who, 
in this undertaking, fell unexpectedly upon the phenomena of aberra- 
tion* and nutation. These admirable discoveries have themselves 
served to show, by the perfect agreement which is thus found to take 

* Subject to be explained hereafter. 



152 ASTRONOMY. 

Chap. XIII. place among observations, that it is hardly to be supposed that the 
annual parallax can amount to 1". The numerous observations of the 
pole star, recently employed in measuring an arc of the meridian 
through France, have been attended with a similar result, as to the 
amount of the annual parallax. From all this we may conclude, that 
as yet there are strong reasons for believing that the annual parallax 
is less than 1", at least with respect to the stars hitherto observed. 

" Thus the semidiameter of the earth's orbit, seen from the nearest 

star, would not appear to subtend an angle of 1'"; and to an observer 

placed at this distance, our sun, with the whole planetary system, would 

occupy a space scarcely exceeding the thickness of a spider's thread. 

Conclusion " If these results do not make known the distance of the stars from 

to be drawn j.jj e ear th, they at least teach us the limit beyond which the stars must 
necessarily be situated. If we conceive a right-angled triangle, having 
for its base half the major axis of the earth's orbit, and for its vertex 
an angle of 1", the distance of this vertex from the earth, or the length 
of the visual ray, will be expressed by 212207, the radius of the earth's 
orbit being unity ; and as this radius contains 23987 times the semidia- 
meter of the earth, it follows that if the annual parallax of a star were 
only 1", its distance from the earth would De equal to 5090209309 radii 
of the earth, or 20086868036404 miles ; that is, more than 20 billions. 
But if the annual parallax is less than 1", the stars are beyond the limit 
which we have assigned. 
Changes " It is evident that the stars undergo considerable changes, since these 

in individual changes are sensible even at the distance at which we are placed. There 
are some which gradually lose their light, as the star <? of Ursa Major. 
Others, as of Cetus, become more brilliant. Finally, there are some 
which have been observed to assume suddenly a new splendor, and then 
gradually fade away. Such was the new star which appeared in 1572, 
A new star, in the constellation Cassiopeia. It became all at. once so brilliant that 
it surpassed the brightest stars, and even Venus and Jupiter when 
nearest the earth. It could be seen at midday. Gradually this great 
brilliancy began to diminish, and the star disappeared in sixteen months 
from the time it was first seen, without having changed its place in the 
heavens. Its color, during this time, suffered great variations. At first 
it was of a dazzling white, like Venus ; then of a reddish yellow, like 
Mars and Aldebaran ; and lastly, of a leaden white, like Saturn. An- 
Another °ther star which appeared suddenly in 1604, in the constellation Ser- 

new star. pentarius, presented similar variations, and disappeared after several 
months. These phenomena seem to indicate vast flames which burst 
forth suddenly in these great bodies. Who knows that our sun may 
not be subject to similar changes, by which great revolutions have 
perhaps taken place in the state of our globe, and are yet to take place. 
Periodical « Some stars, without entirely disappearing, exhibit variations not less 

changes. remarkable. Their light increases and decreases alternately in regular 
periods. They are called for this reason variable stars. Such is the 



FIXED STARS. I53 

star Algol, in the head of Medusa, which has a period of about three Chap. XIII. 
days ; 3 of Cepheus, which has one of five days ; of Lyra, six ; /u. of 
Antinous. seven ; of Cetus, 334 ; and many others. 

" Several attempts have been made to explain these periodical varia- Attempts 
tions. It is supposed that the stars which are subject to them, are, like to explain 
all the other stars, self-luminous bodies, or true suns, turning on their periodical 
axes, and having their surfaces partly covered with dark spots, which chan S es - 
may be supposed to present themselves to us at certain times only, in 
consequence of their rotation. Other astronomers have attempted to 
account for the facts under consideration, by supposing these stars to 
have a form extremely oblate, by which a great difference would take 
place in the light emitted by them under different aspects. Lastly, it 
has been supposed that the effect in question is owing to large opake 
bodies, revolving about these stars, and occasionally intercepting a part 
of their light. Time and the multiplication of observations may per- 
haps decide which of these hypotheses is the true one. 

" One of the best methods of observing these phenomena is to compare Order in 
the stars together, designating them by letters or numbers, and dispos- these obser- 
ing them in the order of their brilliancy. If we find, by observation, vations. 
that this order changes, it is a proof that one of the stars thus com- 
pared, has likewise changed ; and a few trials of this kind will enable us 
to ascertain which it is that has undergone a variation. In this man- 
ner, we can only compare each star with those which are in the neigh- 
borhood, and visible at the same time. But by afterward comparing 
these with others, we can, by a series of intermediate terms, connect 
together the most distant extremes. This method, which is now prac- 
ticed, is far preferable to that of the ancient astronomers, who classed 
the stars after a very vague comparison, according to what they called 
the order of their magnitudes, but which was, in reality, nothing but 
that of their brightness, estimated in a very imperfect manner. 

'•'By comparing the places of some of the fixed stars, as determined Suggestion 
from ancient and modern observations, Dr. Halley discovered that they of'Dr.Halley. 
had a proper motion, which could not arise from parallax, precession, 
or aberration. This remarkable circumstance was afterward noticed 
by Cassini and Le Monnier, and was completely confirmed by Tobias 
Mayer, who compared the places of 80 stars, as determined by Roemer, 
with his own observations, and found that the greater part of them 
had a proper motion. He suggested that the change of place might 
arise from a progressive motion of the sun toward one quarter of the 
heavens ; but as the result of his observation did not accord with his 
theory, he remarks that many centuries must elapse before the true 
cause of this motion could be explained. 

" The probability of a progressive motion of the sun was suggested 
upon theoretical principles by the late Dr. Wilson of Glasgow ; and 
Lalande deduced a similar opinion from the rotatory motion of the sun, 
by supposing, that the same mechanical force which gives it a motion 



4$ 
154 ASTRONOMY. 

Chap. XIII. round its axis, would also displace its center, and give it a motion of 

' ' translation in absolute space 

" If the sun has a motion in absolute space, directed toward any 
quences of ... . 

*uch a the- < l liar t er °* the heavens, it is obvious that the stars in that quarter must 

pyyj appear to recede from each other, while those in the opposite region 

would seem gradually to approach, in the same manner as when walk- 
ing through a forest, the trees toward which we advance are constantly 
separating, while the distance of those which we leave behind is gradu- 
ally contracting. The proper motion of the stars, therefore, in opposite 
regions, as ascertained by a comparison of ancient with modern obser- 
vations, ought to correspond with this hypothesis ; and Sir W. Her- 
schel found, that the greater part of them are nearly in the direction 
which would result from a motion of the sun toward the constellation 
Hercules, or rather to a part of the heavens whose right ascension is 
250° 52' 30", and whose north polar distance is 40° 22'. Klugel found 
the right ascension of this point to be 260°, and Prevost made it 230°, 
with 65° of north polar distance. Sir W. Herschel supposes that the 
motion of the sun, and the solar system, is not slower than that of the 
earth in its orbit, and that it is performed round some distant center. 
The attractive force capable of producing such an effect, he does not 
suppose to be lodged in one large body, but in the center of gravity of 
a cluster of stars, or the common center of gravity of several clusters." 
The following figures, taken from Norton's Astronomy, represent 
the telescopic appearance of some of the double stars. 
Double " There are stars which, when viewed by the naked eye, and even 
and multiple by the help of a telescope of moderate power, have the appearance of 
stars, only a single star ; but, being seen through a good telescope, they are 

found to be double, and in some cases a very marked difference is per- 
ceptible, both as to their brilliancy and the color of their light. These 
Sir W. Herschel supposed to be so near each other, as to obey recipro- 
cally the power of each other's attraction, revolving about their com- 
mon center of gravity, in certain determinate periods. 




Castor, y Leonis, Rigel, Pole Star, crMonoc, ^Cancri. 

Revolutions " The two stars, for example, which form the double star Castor, 
of the multi- have varied in their angular situation more than 45° since they were 
pie stars. observed by Dr. Bradley, in 1759, and appear to perform a retrograde 
revolution in 342 years, in a plane perpendicular to the direction of the 
sun. Sir W. Herschel found them in intermediate angular positions, 
at intermediate times, but never could perceive any change in their 
distance. The retrograde revolution of y in Leo, another double star, 
is supposed to be in a plane considerably inclined to the line in which 
we view it, and to be completed in 1200 years. The stars « of Bootes, 



IXED STARS. 



155 



perform a direct revolution in 1681 years, in a plane oblique to the sun. Chap. Xin. 
The stars £ of Serpens, perform a retrograde revolution in about 375 
years ; and those of y in Virgo in 708 years, without any change of 
their distance. In 1802, the large star £ of Hercules, eclipsed the 
smaller one, though they were separate in 1782. Other stars are sup- 
posed to be united in triple, quadruple, and still more complicated 
systems. 

"With respect to the determination of the real magnitude of the stars, Description 
and their respective distances, we have as yet made but little progress, of nebuke. 
Researches of this kind must be left to future astronomers. It appears, 
however, that the stars are not uniformly distributed through the 
heavens, but collected into groups, each containing many millions of 
stars. We can form some idea of them from those small whitish spots 
called Nebulae, which appear in the heavens as represented in the ac- 
companying illustration. By means of the telescope, we distinguish in 
these collections an almost infinite number of small stars, so near each 
Other, that their 
rays are ordina- 
rily blended by 
irradiation, and 
thus present to 
the eye only a 
faint uniform 
sheet of light. 
Tii at large, 
white, lumi- 
nous track, 
which traverses 
the heavens 
from one pole to 
the other, under 

the name of the Milky Way, is probably nothing but a nebula of this The Milky 
kind, which appears larger than the others, because it is nearer to us. Way a ne- 
With the aid of the telescope we discover in this zone of light such a bula - 
prodigious number of stars that the imagination is bewildered in 
attempting to represent them. Yet from the angular distances of 
these stars, it is certain that the space which separates those which 
seem nearest to each other, is at least a hundred thousand times as great 
as the radius of the earth's orbit. This will give us some idea of the 
immense extent of the group. To what distance then must we with- 
draw, in order that this whole collection may appear as small as the 
other nebulse which we perceive, some of which cannot, by the assist- 
ance of the best telescopes, be made to present anything but a bright 
speck, or a simple mass of light, of the nature of which we are able to 
form some idea only by analogy ? When we attempt, in imagination, 
to fathom this abyss, it is in vain to think of prescribing any limits to 




156 ASTRONOMY. 

Chap. XIII. the universe, and the mind reverts involuntarily to the insignificant 
portion of it which we are destined to occupy. " 
Observa- Before we close this chapter, we think it important to call the atten- 
tions on ta- tj ou f the rea j er t table II , in which will be seen, at a glance (in 
the columns marked annual variation), the general effect of the preces- 
sion of the equinoxes ; and although we have called particular attention 
to the fact elsewhere, we here notice that all the stars, from the 6th to 
the 18th hour of right ascension, have a progressive motion to 
the southward ( — ), and all the stars from the 18th to the 6th hoMX 
of right ascension have a progressive motion to the northward (-f-)j and 
the greatest variations are at h, and 12 h. But these motions are not, in 
reality, the motions of the stars ; they result from motions of the earth. 
Whenever the annual motion of any star does not correspond with this 
common displacement of the equinox, we say the star has a proper 
motion ; and by such discrepancy it has been decided, that those stars 
marked with an asterisk, in the catalogue, have proper motions ; and 
the star 61 Cygni, near the close of the table, has the greatest proper 
motion. 
The paral- From this circumstance, and from the fact of its being a double star, 
lax of 61 it was selected by Bessel as a fit subject for the investigation of stellar 
Cygni disco- p ara n ax . an( j it i s now contended, and in a measure granted, that the 
annual parallax of this star is 0".35, which makes its distance more 
than 592.000 times the radius of the earth's orbit ; a distance that light 
could not traverse in less than nine and one-fourth years. 



PHYSICAL ASTRONOMY. 157 

SECTION III. 
PHYSICAL ASTRONOMY. 



CHAPTER I. 

GENERAL LAWS OE MOTION — THE THEORY OE GRAVITY. 



Chap. I. 



C 1480 In a work like this, designed for elementary in- 

v . J > & J What should 

struction, it cannot be expected that a full investigation of be expe cted 
physical astronomy shall be entered into ; for that subject in this work, 
alone would require volumes ; and to fully appreciate and 
comprehend it, requires the matured philosopher combined 
with the accomplished mathematician. 

We shall give, however, a sufficient amount to impart a good 
general idea of the subject — if one or two points are taken 
on trust. 

For elementary principles we must turn a moment to natu- Elementary 
ral philosophy, and consider the laws of inertia, motion, and principles. 
force. Motion is a change of place in relation to other bodies 
which we conceive to be at rest ; and the extent of change in 
the time taken for unity is called velocity, and the essential 
cause of motion we denominate force. 

A double force will give a double velocity to bodies moving Velocity the 

pi' •! • ... t , • 7 measure of 

freely in void space, or in an unresisting medium — a triple 
force, a triple velocity, &c. This is taken as an axiom — and 
hence, when we consider mere material points in motion, the 
relative velocities measure the relative amounts of force. 

There are three elements to motion, which the philosopher 
never loses sight of; or we may say that he never thinks of 
motion without the three distinct elements of time, velocity, and 
distance, coming into his mind. 

Algebraically, we put t, v, and d, to represent the three ele- 
ments, and then we have this important and general equation, 

tv=d (1) 

N 



158 ASTRONOMY. 

Chap. I. , d d 

From this we derive v=— (2) and t=- ('3) 

Expression t J V v ' 

( 149.) As forces are in proportion to velocities (when mo- 
mentum is not in question ), therefore, if we put / and F to 
represent two forces corresponding to the distances d and D, 
which are described in the times t and T, then by making use 
of equation ( 2 ), in place of the velocities, we have 

f:F::* ( :§ (4)* 

The law of ( 150. ) A body at rest, has no power to put itself in mo- 
tion, and having no self power, no internal force or will, in 
any shape, it cannot increase or diminish the motion it may 
have, or change the direction it may be moving. This is the 
law of inertia. It cannot of itself change its state ; and if it 
is changed it must be acted upon by some external force; 
and this accords with universal experience ; and this law is 
the most natural and simple of any we can imagine, but it is 
only in the motion of the heavenly bodies that it is fully 
exemplified. 
Some central The earth, moon, and planets move in curves — not in 
force must pj-j^ l mes> The directions of their motions are changed. 

act on the ° 

motions of Something external from them must, therefore, change them ; 
the earth, f 0T £k e ] aw f i neT ii a would continue a motion once obtained 

moon, and . .,,. •»-,- i • •> • j • i • i 

planets. m a straight line. JNow this force must exist within the or- 
bit of every curve; we therefore naturally refer it to the 
body round which others circulate. The earth and planets 
go round the sun, and if we could suppose a force residing in 
the sun to extend throughout the system sufficient to draw 
bodies to it, this would at once account not only for the 
planets deviating from a right line, but would account for a 
constant deviation of all bodies to that point, and the preser- 
vation of the system. 

The moon's The moon goes round the earth, constantly deviating from 
the tangent of its orbit, and the law of inertia is constantly 



motion con 
sidered. 



* We number the proportions the same as equations, for a propor- 
tion is but an equation in another form. 



THE EARTH'S ATTRACTION. 159 

urging it to rise from the center ; the two on an average balan- Chap. i. 
cing each other, retains the moon in an orbit about the 
earth. 

Now what and where is this force ? Is it around the 
earth, or within the earth ? Is it electrical or magnetic ? or 
fe it that same force ( call it what we may ) that makes a 
body fall toward the earth's center when unsupported on a 
resting base ? 

A trifling incident, the fall of an apple from a tree, seems contempia. 
to have led the mind of Newton to the contemplation of this t»«w of gir 
force which compels and causes bodies to fall, and he at once t ^ ac 
conceived this force to extend to the moon and to cause it to 
deviate from the tangent of its orbit. 

The next consideration was, whether if this were the force, 
it was the same at the distance of the moon, as on the sur- 
face of the earth ; or if it extended with a diminished amount, 
what was the law of diminution ? 

Newton now resorted to computation, and for a test he incipient 
conceived the force in question to extend to the moon, undi- s , teps t0 l I 

x theory 01 

minished by the distance ; and corresponding thereto he de- gravity. 
cided that the moon must then make a revolution in its orbit 
in 10 h. 55 m. But the actual time is 27 d. 7h. 43 m., 
which shows that if the force is the same which pervades a 
falling body on the surface of the earth, it must be greatly 
diminished. 

Now by making a reverse computation, taking the actual important 
time of revolution, and finding how far the moon did really com P uta - 
fall from the tangent of its orbit in one second of time, it was 
found to be about 3 F Vo P ar * °f 16 jj feet — the distance a 
body falls the first second of time. 

But the distance to the moon is about 60 times the radius 
of the earth, and the inverse square of this is ggV o» which 
corresponds to the actual fall of the moon in one second. 

(151.) It is a well-established fact in philosophy, and a principle 
geometrically demonstrated, that any force or influence exist- m P hllos °P h ? 
ing at a point, must diminish as it spreads over a larger 
space, and in proportion to the increase of space. But space 
increases as the square of linear distance, as we see by Fig. 28. 



160 ASTRONOMY. 

Chap - t- A double distance spreads the influence over four times the 
space, whatever that influence may be ; a triple distance, nine 
times the space, etc., the space increasing as the square of 

Fig. 28. 




the distance. Therefore, any influence spreading in all di- 
rections from its central point must be enfeebled as the square 
of the distance. 
The theory Erom observations and considerations like these, Newton 
gravity. established the all-important and now universally admitted 
theory of gravity. 

This theory may be summarily stated in the following 
words : 

Every body of matter in the universe attracts every other body, 
in direct "proportion to its mass, and in the inverse proportion to 
the square of the distance. 
This theory Some attempts have been made, from time to time, to call 
well estab- ^he truth of this theory in question, and substitute in its 
place the influence of light, caloric, and electricity; but any 
thing like a close application shows how feebly all such sub- 
stitutes stand the test. 

The theory of gravity so exactly accounts for all the phy- 
sical phenomena of the solar system, that it is impossible it 
should be false ; and although we cannot determine its nature 
or its essence, it is as unreasonable to doubt its existence, as 
to doubt the existence of animate beings, because we know 
nothing of the principle of life. 
Attraction (152.) According to the theory of gravity, every particle 
composing a body has its influence, and a very irregular body 
may be divided in imagination into many smaller bodies, and 
the center of gravity of each taken as the point of attraction, 
and all the forces resolved into one will be the attraction of 
the whole body. 



of an irregu 
lar body 



STANDARD OF FORCE. 161 

In a sphere eomposed of homogeneous particles, the aggre- chap. i. 
gate attraction of all of thern will be the same as if all were , Vttraction of 
compressed at the center; but this will be true of no other a sphere, 
body. The earth is not a perfect sphere, and two lines of 
attraction from distant points on its surface may not, yea, 
will not, cross each other at the earth's center of gravity. 
( See Fig. 10.) 

(153.) A particle anywhere inside of a spherical shell of Attraction 
equal thickness and density, is attracted every way alike, and inside of a 
of course would show no indication of being attracted at all. ^ e] e " ca 
Hence a body below the surface of the earth, as in a deep pit 
or well, will be less attracted than on the surface, as it will 
be attracted only by the diminished sphere below it. At the 
center of the earth a body would be attracted by the earth the cen 1 ter f 
every way alike, and there would be no unbalanced force, a sphere. 
and of course no perceptible or sensible attraction.* 

( 154 .) The attractive power on the surface of any perfect Ex ress . 
and homogeneous sphere may be expressed by the mass of the for the at- 
sphere divided by the square of the radius. traction on 

r J z J the surface of 

Consider the earth a sphere (as it is very nearly), and a sphere, 
put E to represent its mass, and r its mean radius, then 

E 

— = g — 16-^ feet. 

E 

This attractive force, algebraically expressed by j-> we call g, 

and it is sufficient to cause bodies to fall lGyL- feet during 
the first second of time. If the earth had contained more 
matter, bodies would have fallen more than 16-jL feet the 
first second; if less, a less distance. 

With the same matter, but more compact, so that r? would The definite 

ii '17-ri ■& ill ii attraction of 

be less with E the same, — would be greater, and the attrac- t he earth. 

tive power at the surface greater, and bodies would then fall 
more than 16^ feet the first second of their fall. 

Now we say this 16 T \ feet is the measure of the earth's 
attraction at its surface, and it is made the unit and standard 
measure, directly or indirectly, for all astronomical forces. 

* See Robinson's Natural Philosophy, page 16. 
11 ' N* 



162 ASTRONOMY. 

Chap. i. For this reason, we call the undivided attention to this 

force, the known — the noted — the all-important IQ^feet. 
To find the ( 155. ) By the theory of gravity, we can readily obtain an 
rac ion o ana iyti ca l expression for the attraction of a sphere at anv dis- 

a sphere at J L x. J 

any distance, tance from the center, after knowing the attraction at the 
surface. For example. Find the value of the attraction of 
the earth, at the distance of D from its center ; r being the 
radius of the earth, and g the gravity at the surface ; put x 
to represent the attraction sought. Then by the theory, 

9 '* -'• ~ 2 : giS 0r > x = v(^0 ( 5 ) 

As g and r are constant quantities, the variations to x will 
correspond entirely to the variations of D 2 . We shall often 
refer to this equation. 
Anexpres- (156.) As every particle of matter in the universe at- 
sion for the tracts every other particle, therefore the moon attracts the 
traction of eartn as weu * as tne earth attracts the moon ; and the extent 
two bodies, by which they will draw together, depends on their mutual at- 
traction. If m represents the mass of the moon, and R the 
radius of the lunar orbit ; then, 

E 

The earth will attract the moon by the force -^. 



m 



The moon will attract the earth by the force -=^ 

E-\~m 
The two bodies will draw together by the force ^ 2 . 

If we substitute the value of g, as found in ( 154), in equa- 

. E 

tion (5 ), and making H = J), then we have the expression — • 

The spirit of these expressions will be more apparent when 
we make some practical applications of them, as we intend 
soon to do. 



KEPLER'S LAWS. 



163 



CHAPTER II. 

kepler's laws — demonstration of the second and third — 
how a planetary body will find its orbit. 



( 157. ) In this chapter we design to make some examina- chap. h. 
tion of Kepler's laws, recapitulating them in order. Examina- 

The orbits of the planets are ellipses, having the sun at tionsofKep- 

j. , 7 . -. . ler's laws. 

one of their joa. 

This law is but a concise statement of an observed fact, 
which never could have been drawn from any other source 
than observation ; but the second law, namely, 

That the radius vector of any planet ( conceived to be in mo v 
tion ) sweeps over equal areas in equal times is susceptible of 
a rigid mathematical demonstration, under the following gen- 
eral theorem. 

Any body, being in motion, and constantly urged toivard any A s eneral 
fixed point, not in a line with its motion, must describe equal 
areas in equal times round that point. 

Let a moving 

U J l. i. A Fi S- 29 « 

body be at A, 
having a veloci- 
ty which would 
carry it to B, 
say in one sec- 
ond of time. By 
the law of iner- 
tia, it would 
move from B to 
C, an equal dis- 
tance, in the next second of time. But during this second 
interval of time, let us suppose it must obey an impulse or 
force from the pcint S, sufficient to carry it to D, It must 
then, by the composition of forces explained in natural phi- 
losophy, describe the diagonal B E, of the parallelogram 
BDEQ. 




Its demon- 
stration. 



verse of the 
theorem. 



164 ASTRONOMY. 

Chap. ii. Now in the first interval of time, we supposed the moving 
body described the triangle SAB. The second interval, it 
would have described the triangle S B C, if undisturbed by 
any force at S, but by such a force it describes the triangle 
S B E; but the triangle S B E, is equal to the triangle 
SBC, because they have the same base S B, and lie between 
the parallels S B and E C. Also the triangle S B C is 
equal to the triangle SAB, because they terminate in the 
same point S, and have equal bases, A B and B C. There- 
fore the triangle S A B is equal to the triangle S B E, be- 
cause they are both equal to the triangle SBC; that is, the 
moving body describes equal areas in equal times about the 
point Sj and this is entirely independent of the nature of the 
force at S; it may be directly or inversely as the distance, or 
as the square of the distance. 
The con- The converse of this theorem is, that when a body describes 
equal areas in equal times round any point, the body is con- 
stantly urged toward that point, and therefore as the planets 
are observed to describe equal areas in equal times round the 
sun, their tendency is toward the sun, and not toward any 
other point within the orbits. 
Kepler's (158.) The third law of Kepler is most important of all, 

thud law name }y — I'he squares of the times of revolution are to each 

proves that .. . -r> i • i 

the sun's at- °t" £r as the cubes of the distances from the sun. By this law 
traction is it is proved, that it is the same force which urges all the 
inverse y as ' pi ane ^ s ^ ^ e same point, and that its intensity is inversely as 
the distance, the square of the distance from that point ( the center of the 
sun ), confirming the Newtonian theory of gravity. 

To show this, let us suppose that the 
planets revolve round the sun in circular 
orbits (which is not far from the truth), 
and let P ( Fig. 30 ) represent the posi- 
tion of a planet ; F the distance which 
the planet is drawn from a tangent during 
unity of time ; in the same time that it 
describes the indefinite small arc c ; and 
the number of times that c is contained in the whole circum- 
ference, so many units of time, then, must be in one revolution. 




SOLAR SYSTEM. 165 

If D is the diameter of the orbit and t the time of revolu- Chap, h. 
tion, then will 

t=~, ... (1) 

So for any other planet. If / is the force urging it toward An impor- 
the sun, a its corresponding arc, Tits time of revolution, and tant troth de- 

t-»i t i • 1 monstrated. 

B the radius of its orbit ; then, reasoning as before, 

2*-B 

t=A-t, • • • (2) 

a 
By comparing ( 1 ) and ( 2 ) we have 

c a 

_ D 2 AB 2 

By squaring, t 2 : T 2 :: — : — — . 

By Kepler's law, t 2 : T 2 : : r 3 : B*. 

By comparing the two last proportions, and observing that 
2r may be put for D, and reducing, we have 

c 2 a 2 

But by the well-known property of the circle, we have 

F : c :: c : 2r; or, c 2 = 2rF. 

In like manner, . . . a 2 = 2 Bf. 

Substituting these values in the last proportion, and redu- 
cing, we have 

LI ■.*•.> 



Or, . . Bf : r#:: r : .£. 

Hence, . B 2 f=r 2 F; or, F : f :: B 2 

Or, J 7 : / 



A* 2 

1 



r 2 * i? 2 



That is ; the attractive force of the sun is reciprocally pro- 
portional to the square of the distance. 

( 159.) If we commence with the hypothesis, that bodies The ^e^ 

tend toward a central point with a force inversely propor- of gravity 



166 ASTRONOMY. 

Ckap - n - tional to the squares of their distances, and then compute 
and laws of the corresponding times of revolution, we shall find that the 
in Kepler's squares of the times must be as the cubes of the distances. Hence 
third law. Kepler' s third law is but the natural mathematical relation 
which must exist between times and distances among bodies 
moving freely, in circular orbits, animated by one central 
force which varies as the inverse square of the distance. 
An inquiry. ( 160. ) Having shown that Kepler's third law is but a 
mathematical theorem when the planets move in circles and 
their masses inappreciable in comparison to that of the sun's, 
we now inquire whether the law is true, or only approximately 
true, when the orbits are ellipses, and their masses consid- 
erable. 
How answer- On one of these points of inquiry, the reader must take our 
ed - assertion; for its demonstration requires the use of the inte- 

gral calculus, a subject that we designed not to employ in this 
work. Kepler's third law supposes all the force to be in the 
central body, and the planets only moving points. But we 
have seen in Art. ( 120 ) that the attracting force on any 
planet is the mass of both sun and planet divided by the 
square of their mutual distance; and therefore when the 
mass of the planet is appreciable, the force is increased, and 
Masses of the time of revolution a little shortened. But the fact that 
the planets K e pi e r's law corresponds so well with other observations 
compared to proves that the masses of all the planets are inappreciable 
the sun. compared to the mass of the sun. 

Kepler's ( 161. ) As to the other point, we state distinctly that the 
third law ma- planets ( considered as bodies without masses) revolving in 
thematioaiiy e ]j- seg f evcr g0 grea t eccentricity, the squares of the times 

true in el lip- *- ° , , 

tic orhits. of revolution are to each other as the cubes of half the greater 
axes of the orbits. 

We shall not attempt a demonstration of this truth ; but 
hope the following explanation will give the reader a clear 
view of the subject. 

Bodies revolving in ellipses round one of the foci, may be 
considered to have a rising and a falling motion; something 
like the motion of a pendulum. The motion of a pendulum 
depends on the force of gravity, the length of the pendulum, 



PLANETARY MOTION. 167 

and the distance the pendulum was first drawn aside. The chap, u. 

motion of a planet depends on the force of gravity, its mean 

distance from the sun, and the original impulse first given to a common 

it. Most persons, who have not investigated this subject, error of opin- 

imagine that each planet must originally have had precisely 

the impulse it did have to maintain itself in its orbit; and so 

it must, to maintain itself in just that definite orbit in which 

it moves. But had the original impulse heen different, either as 

to amount or direction, or as to both, then by the action of gravity 

and inertia, the planet would have found a corresponding orbit. 

(162.) The force of gravity, from the action of any attract- Examin- 
ing body, is always as the mass of the body divided by the square tion of the 
of its distance. Algebraically, if M is the mass of the body, motions j n 
r its distance, and F the force at that distance, then (see 118) elliptic orbits 

we have - - - ^-=F. (See Fig. 28.) 

Now if the planet has such a velocity, c, as to correspond 
with the proportion F : c : : c : 2r, 



Or, - 



f'2M 

:= t J2rF=-^ , and that velocity at 



right angles to r (Fig. 28), then the planet's orbit would be a 
circle, with the radius r. If the velocity had been less in 
amount than this expression, and still at right angles to r, then 
the planet would fall within the circle, and the action of gra- 
vity would increase the motion of the planet ; and the motion 
would increase faster than the increased action of gravity : 
there would be a point, then, where the motion would be sufficient further from * 
to maintain the planet in a circle, at its then distance ; but the the sun. Be- 
direction of the motion will not permit the planet to run into ^ ^ "' 
the circle, and it must fall within it. 

The motion continues to increase until its position becomes 
at right angles to the radius vector ; the motion is then as 
much more than sufficient to maintain the planet in a circle, 
as it was insufficient in the first instance; it therefore rises, 
by the law of inertia, and returns to the original point P, 
where it will have the same velocity as before ; and thus the 
planet vibrates between two extreme distances. 
12 



168 



ASTRONOMY. 



Chap. II 

Gravity and 
original ve- 
locity deter- 
mine the ec- 
centricityand 
mean distan- 
ces of the or- 
bits. 



A hypothe- 
tical case. 



How a 
planet finds 
its orbit. 



If the velocity, on starting from the point P, were very 
much less than sufficient to maintain a circle, at that distance, 
then the orbit it would take would be very eccentric, and 
its mean distance much less than r. If the original velocity 
at P were greater than to maintain it in a circle, it would 
pass outside of this circle, and the point P would be the peri- 
helion point of the orbit. 

Thus, we perceive, that the eccentricity of orbits and mean 
distances from the sun, depend on the amount and direction 
of the original impulse, or velocity which the planet has in 
some way obtained; and it is not necessary that the planet 
should have any definite impulse, either in amount or direction, to 
move in an orbit, if the direction is not directly to or from the sun. 
(163.) For a more definite explanation of this subject, let 
us conceive a planet launched out into space with a velocity 
sufficient to maintain it in a circle at the distance it then hap- 
pened to be, but the direction of such velocity not at right 
angles to the sun, then the orbit will be elliptical, and the 
degree of eccentricity will depend on the direction of the 
motion ; but the longer axis of the orbit will be equal to the 
diameter of the circle, to which its velocity corresponds ; and 

the time of its revolution will be 
the same, whether the orbit is 
circular or more or less elliptical. 
Let P (Fig. 31) be the posi- 
tion of a planet, S the sun ; and 
let the velocity, a, be just suffi- 
cient to maintain the planet in 
a circle, if it were at right angles 
to 8 P. 

Now to find the orbit that this 
planet would describe, draw the 
line P C at right angles to a, 
and from S let fall a perpendi- 
cular on PC; SC will be the 
eccentricity of the orbit, and PC 
will be the half of its conjugate 
axis ; and with these lines the whole orbit is known. 



Fig. 31. 




PLANETARY MOTION. 



169 



( 164.) Now let us suppose that a planet is rather carelessly chap, ii . 
launched into space, with a velocity neither at right angles to 
the sun, nor of sufficient amount to maintain it in a circle, at 
that distance from the sun. 



Fig. 32. 




will find their 
orbits, what- 
ever be the 
direction and 
force of their 
original mo- 
tion. 



Let P (Fig. 32) represent the 
position of the planet, a the 
amount and direction of its hap- 
hazard velocity during the first 
unit of time. The direction of 
the motion being within a right 
angle to S P, the action of gra- 
vity increases 
the velocity ^./*-\ 
of the planet, ^ 
on the same 

principle that a falling body in- 
creases in velocity ; and the planet 
goes on in a curve,describing equal 
areas in equal times round the point 
S; and it will find a point, p, where 
its increased velocity will be just 

equal to the velocity in a circle whose radius is the diminished 
distance S p. From the point p, and at right angles to a, 
draw p C, &c, forming the right angled triangle p C S. S C 
is the eccentricity, S a the mean distance, and p C half the 
conjugate axis of the orbit. 

If the planet is launched into space in the other direction, 
the action of gravity will diminish its motion, and will bring w \ . e sym " 

D » ' ° metrical on 

it at right angles to the line joining the sun; it is then at its each side of 
apogee, with a motion too feeble to maintain a circle at that a P°s ee ; 
distance; and it will, of course, approach nearer and nearer 
to the sun by the same laws of motion and force that it receded 
from the sun ; hence the curve on each side of the apogee 
will be symmetrical ; and the same reasoning will apply to the 
curve on each side of the perigee ; and, in short, we shall 
have an ellipse. 

To sum up the whole matter, it is found by a strict exami- t 

x ' •* tant conclu- 

nation of the laws of gravity, motion, and inertia, that whatever sion. 



The orbits 



perigee. 



An impor- 



170 



ASTRONOMY. 



Chap. ii. may be the primary force and direction given to a planetary 
body ( if not directly to or from the sun ), the planet will find 
a corresponding orbit, of a greater or less eccentricity, and of a 
greater or less mean distance ; and whatever be the eccentricity 
of the orbit, the real velocity, at the extremity of the shorter axis, 
will be just sufficient to maintain the planet in a circular orbit, at 
that mean distance from -the sun* 

* Let S be the sun, and P the position of a planet as repre- 
sented in the annexed figure, and we may now suppose it to 
the asteroids burst into fragments, the figure representing three fragments 
only ; the velocity and direction of one represented by a ; of 
another by b, and of a third by c, &c. 

Fig. 33. 



Theory of 
Dr. Olbers 
concerning 




As action is just equal to reaction, under all circumstances, 

therefore the bursting of a planet can give the whole mass no 

additional velocity ; a small mass may be blown off at a great 

velocity, but there will be an equal reaction on other masses, 

On the « n ^ e pp 0S ite direction. 

bursting of a L L 

planet, the The whole might simply burst into about equal parts, and 
fragments then they would but separate, and all the parts move along 
orbits corre- m * ne same general direction, and with the same aggregate 
sponding to velocity as the original planet. The bursting of a rocket is 
tieTanT 0C1 a ver y mmute > ou * a ver y faithful representation of such an 
tions. explosion. 



KEPLER'S LAWS. 171 

( 165.) To see whether Kepler's third law applies to ellipses, Chap. n. 
we represent half the greater axis of any ellipse by A, and Kepler's 
half the shorter axis by B, and then (3.1416) AS is the area third law ri - 
of the ellipse. Also, let a represent the velocity or distance ^ 0T0US J me 

-!• - 1 - •> in relation to 

~ ellipses, as 

If the velocities of the several fragments were equal, the well as to 
times of their revolutions would be equal ; but the eccentri- circles - 
cities of the several orbits would depend on the angles of a, 
b, c, &c, with S P. If a is at right angles to S P, and just 
sufficient to maintain the planet in a circle at that distance, 
then its orbit would have no eccentricity. If still at right 
angles, but not sufficient to maintain a circle at that distance, 
then S P would be the greatest radius of the orbit. Hence, 
we perceive, there is an abundance of room to have a multi- 
tude of orbits passing through the same point, during the 
first one or two revolutions ; and the times of such revolu- 
tions may be equal, or very unequal. In short, there is no 
physical impossibility to be urged against the theory of Dr. 
Olbers, that the asteroids are but fragments of a planet. 

The objection is (if an objection it can be called) that 
these planets have not, in fact, a common node, nor have an 
approximation to one ; nor have they an approximation to a 
common radius vector, as S P. But the objection vanishes 
when we consider that the elements of the different orbits 
must be variable ; and time, a sufficient length of time, would 
separate the nodes and change the positions of the orbits so 
as to hide the common origin, as is now the case. 

But if it be true that these planets once had a common 
origin in one large planet, it is possible to find the variable 
nature of the elements of their orbits to such a degree of 
exactness as to trace them back to that origin — define the 
place where, and the time when, the separation must have 
occurred. 

If, however, a planet should burst at one time, and after- 
ward one or more of the fragments burst, there could be no 
tracing to a common origin ; hence it is possible that the 
asteroids in question may have a common origin, and it be 
wholly beyond the power of man to show it. 



172 ASTRONOMY. 

Chap. ii. that the planet will move in a unit of time, when at the ex- 
tremity of its shorter axis ; then \aB will express the area 
described in that unit of time. 

But as equal areas are described in equal times, as often 
as this area is contained in the whole ellipse will be the num- 
ber of such units in a revolution. Put 1= that number, or 
the time of revolution ; then 

(3.1416)45 2(3.1416)4 



t: 



\aB a 



Let A' and B' be the semiaxes of any other ellipse ; a' the 
velocity at the extremity of B', and f the time of revolution ; 

, 2.(3.1416)4' 

then will - ■ t'——± —+ — . 

a 

By comparing these equations, and rejecting common fac- 

A A' 

tors, we have - t : t' : : — : — -. 



I2M I2M 

But by Art. 162, a=^|— , and a'=^j— - 



M mass of sun) ; and putting the values of a and a, in the 
above proportion, we have 

i JA A' J A' 

t : t' : : A v : — g — ; 

J2M J2M 

Or, - - t : t' :: AjA : A! JA! . 

By squaring t 2 : t' 2 :: A 2 : A' 3 ; which is 

Kepler's third law. 
Eccentrici- (166.) We have seen, in articles 126 and 127, that the 
ties o t e eccen t r i c ity of an orbit depends on the direction of the motion 

planetary or- . - 1 

bits change *° * ne radius vector, when the planet is at mean distance. If 
by their mu- that direction is at right angles to the radius vector, at that 
tions " time, then the eccentricity is nothing. If its direction is very 
acute, then the eccentricity is very great, &c. 

Now suppose another planet to be situated at B (Fig. 30) ; 
its attraction on the planet, passing along in the orbit p a, is 
to give the velocity, a, a direction more at right angles to 



KEPLER'S LAWS. 173 

Sp, and thus to diminish the eccentricity of the orbit. If Chap. h. 
the disturbing body, J5, were anywhere near the line C S, its The mean 
tendency would be to increase the eccentricity; and thus, in dlstan cesne- 
general, A disturbing body near a line of the shorter axis of 
an orbit, has a tendency to diminish the eccentricity of the orbit 
of the disturbed body ; and,, anywhere near a line of the greater 
axis, has a tendency to increase the eccentricity. Hence the 
eccentricities of the planets change in consequence of their 
mutual attractions; but their mean distances never change. 

(167.) As the time of revolution is always the same for 
the same mean distance, whatever be the eccentricity of the 
orbit, therefore if we conceive a planet to turn into an infi- 
nitely eccentric orbit, and fall directly to the sun, the time of 
such fall would be half a revolution, in an orbit of half its 
present mean distance, as we perceive, by inspecting Fig. 34. 

Hence, by Kepler's third law, we can compute the „. „ . The prm- 
time that would be required for any planet to fall to / r\ ciples and 

i T , . / \ the computa- 

tne sun. .Let x represent the time a planet would / \ tion of the 
revolve in this new and infinitely eccentric orbit ; then, 

by Kepler's law, 

t 2 
t 2 : x 2 :: 2 3 : l 3 , or, x 2 =—. 

8 

Therefore half of the revolution, or simply the time 
of the fall, must be expressed by -, or, 



time required 
for the plan- 
ets to fall to 
the sun. 



2^/8 V2 

that is, to find the time in which any planet would 
fall to the sun, if simply abandoned to its gravity, or the time 
in which any secondary planet would fall to its primary, divide 
its time of revolution by four times the square root of two. 

By applying this rule, we find that 

Days. 

Mercury would fall to the sun in 15 

Venus, 39 

Earth, 64 

Mars, 121 

Jupiter, 765 

Saturn, 1901 

Uranus, 5424 

The moon would fall to the earth in 4d. 19 h. 54 m. 36 s. 



h. 


m. 


13 


13 


17 


19 


13 


39 


10 


36 


21 


36 


23 


24 


16 


52 



174 ASTRONOMY. 



CHAPTER III. 

MASSES OF THE PLANETS DENSITIES PRESSURE ON THEIR 

SURFACES. 

Chap. in. (168.) If the earth contained more matter, it would 
„. attract with greater force; and if the sun has a greater 

Masses mea- ° ° 

sured by at- power of attraction than the earth, it is because it contains 
traction. more matter than the earth ; and therefore, if we can find the 
relative degree of attraction between two bodies, we have 
their relative masses of matter. 

If the earth and sun have the same amount of matter, they 
will attract equally at equal distances. Let M be the mass 
of the sun, and E the mass of the earth, then ( at the same 
unit of distance), the attraction of the sun is, to the attraction of 
the earth, as Mto E. 

But attraction is inversely as the square of the distance. 

M 
Hence the attraction of the sun at D distance, is -=— ; and 

E 

the attraction of the earth at R distance is -^. 

Gravity of The earth is made to deviate from a tangent of its orbit 
the sun is ^y fo Q attraction of the sun ; and the moon is made to deviate 

measured by « • "• -i • ' i i • 

the devia- from a tangent of its orbit by the attraction of the earth, and 
tion of the the amount of these deviations will give the respective 

earth from a , f> t t • . • i 

tangent of its amounts oi solar and terrestrial gravity. 

orbit. If we take any small period of time, as a minute or a sec- 

ond, and compute the versed sine of the arc which the earth 
describes in its orbit during that time, such a quantity will 
express the sun's attraction; and if we compute the versed 
sine of the arc which the moon describes in the same time, 
that quantity will express the attraction of the earth. 

How to com- j n Fig ure 30, Art. 158, ^represents the versed sine of an 
arc ; and if we take D to represent the mean distance be- 



pute the com 
parative 



masses of the tween the earth and sun, and consider the orbit a circle 
sun and earth ^ ag we ma y ^hout error, 164), the whole circumference is 



MASSES OF THE PLANETS. 175 

vD (*r= 6.2832). Divide the whole circumference by the Chap. m. 
number of minutes in a revolution ; say T, and the quotient 
will represent the arc a (Fig. 30). When T is very small, 
and of course a very small, the chord and arc practically coin- 
cide ; and by the well known property of the circle, we have 

2D : a :: a : F>, Or, ^=|g, . (1) 

t> j *$'/. „ " 2 D 2 ' a 2 Dn 

±Jut a = —==-'. hence, a 2 = — =r— - -, and ^-^ = 



T ' ' ^ 2 ' 2i> 2T 2 

2T- 



That is, F = „ ; which is an expression for the sun's 



attraction at the distance of the earth. But -=- - is also an 

D 2 

expression for the sun's attraction at the same distance ; 
therefore, _ = ___; Or, M=^. 

In the same manner, if R represents the radius of the lunar 
orbit; t the number of minutes in the revolution of the 
moon ; the mass of the central attracting body ( in this case 
the earth ) must be expressed by 

?r 2 Rz 



JE = 



2P 



7?3 Ti 3 

Therefore, E : M : : — : j,-. 

This proportion gives a relation between the masses of the 
earth and sun expressed in known quantities. 

If we assume unity for the mass of the earth, we shall 
have for the mass of the sun, 

t 2 D 3 
M =T^' ■ ■ ■ ^ 

(169.) This is a very general equation, for D may repre- The general 
sent the radius of the earth's orbit, or the orbit of Jupiter or application 
Saturn, and T will be the corresponding time of revolution. tion 1S equa " 
Also R may represent the radius of the lunar orbit, or the 



176 ASTRONOMY. 

chap. hi. orbit of one of Jupiter's or Saturn's moons, and then t will 

be its corresponding time of revolution. 
The results This equation, however, is not one of strict accuracy, as 
of the equa- ^ e distance a planet falls from the tangent of its orbit,, in a 

tion will not 

accurate, and definite moment of time, is not, accurately ■=— , but — — -— 

why ? 

( see 156 ), E being the mass of the planet. The force 
which retains a moon in its orbit is not only the attracting 
mass of the central body, but that of the moon also. But 
the planets being very small in relation to the sun, and in 
general the masses of satellites being very small in respect to 
their primaries, the errors in using this equation will in gen- 
eral be very small. The error will be greatest in obtaining 

Corrections J ..' .. ' • ' P . . , 

for equation the mass of the earth, as in that case the equation involves 
( A) - the periodic time of the moon ; which period is different from 

what it would be were the moon governed by the attraction 
of the earth alone ; but the mass of the moon is no inconsid- 
erable part of the entire mass of both earth and moon; and 
also the attraction of the sun on the combined mass of the 
earth and moon, prolongs the moon's periodical time by about 
its 179th part. 

With these corrections the equation will give the mass of 
the sun to a great degree of accuracy ; but we can determine 
the mass of the sun by the following method : 
a more ac- From Art. 155, we learn that the attraction of the earth 

curate equa- > „ \ 

t,on - at the distance to the sun, is 9\-f^)- 

By Art. 168, we have just seen that the attraction of the 

n 2 D 

sun on the earth, is ^~^ ; therefore, 

AJL 

E: if ::i ,_ : _. 

Taking the mass of the earth as unity, we have 

¥=^r>- ■ - (jB) 

Equation ( B ) is more accurate than equation ( A ), 



MASSES OF THE PLANETS. 



177 



because ( B) does not involve the periodical revolution of the Chap. ni. 
moon, which requires correction to free it from the effects of 
the sun's attraction. To obtain a numerical expression for How to ot>- 
the mass of the sun. M, the numerator and denominator of the me r ica] e n r g 
right hand member of equation ( B ), must be rendered homo- suit, 
geneous ; and as g, the force of gravity of the earth, is ex- 
pressed in feet ( corresponding to T in seconds ), therefore r 
the mean radius of the earth, and D the distance to the sun, 
must be expressed in feet. But from the sun's horizontal 
parallax, we have the ratio between r and D ( see 127 ), 
which gives D = 23984?-. 



This reduces the fraction to 



?r 2 (23984) 3 r 
W 2 ' 



But to ex- 



press the whole in numbers, we must give each symbol its 
value ; that is, «■ == 6.2832 ; r = ( 3956 ) ( 5280 );g= 16.1 ; 
T= 31558150, the number of seconds in a sidereal year. 
(6.2832) 2 (23984)3(3956)(5280) 



Therefore, 



M= 



(32.2)(31558150>' 



It would be too tedious to carry this out, arithmetically, An example 
without the aid of logarithms, and accordingly we give the showin g the 

.,,., great utility 

loganthmetical solution, thus, of logarithms 



6 .2832 log. 0.798178x2 
23 .984 log. 4.380000x3 

3956 log. 

5280 log. 
Logarithm of the numerator, 



1 .596356 

13 .140000 

3 .597256 

3 .722632 

22 .056244 



32.2 log 

31558150 log. 7.499114x2 

Logarithm of the denominator, 
Therefore M= 354945, whose log. is 



1 .507856 The mass of 

14 .998228 the sun de " 



16 .506084 
5 .550160 



termined. 



That is, the mass or force of attraction in the sun is 
354945 times the mass or attraction of the earth. La Place , 
12 



178 ASTRONOMY. 

Chap. in. says it is 354936 times ; but the difference is of no conse- 
quence. 

Equation ( A ) gives 350750, but equation ( £ ), as we 
have before remarked, is far more accurate, and the result 
here given, agrees, within a few units, with the best author- 
ities. 

Equation (B) is not general; it will only apply to the 
relative masses of sun and moon, because we do not know 
the element g, the attraction, on the surface of any other 
planet, except the earth. That is, we do not know it as a 
primary fact ; we can deduce it after we shall have determined 
the mass of a planet. 

Equation ( A) is general, and although not accurate, when 
applied to the earth and sun, is sufficiently so when applied 
to finding the masses of Jupiter, Saturn, or Uranus ; because 
these planets are so remote from the sun, that the revolutions 
of their satellites are not troubled by the sun's attraction. 
To find the ( 170. ) To find the mass of Jupiter ( or which is the 
masses o u- game thing, the mass of the sun when Jupiter is taken as 

piter, Saturn, ° x 

and Uranus, unity), we conceive the earth to be a moon revolving about the 
sun, and compare it with one of Jupiter's satellites revolving 
round that body. To apply equation (^4), let the radius of the 
earth equal unity, then the radius of Jupiter must be 11.11 
(Art. 131 ); and as observation shows the radius of Jupiter's 
4th satellite is 26.9983 times its equatorial radius, therefore 
the distance from the center of Jupiter to the orbit of its 
4th satellite, must be the following product (11.11) (26.9983), 
which corresponds to R in the equation. D = 23984; 
T= 365.256; ^= 16.6888. 

t 2 D 3 
Therefore, by applying equation ( A ), ( M == ) ; we 

(16.6888) 2 (23984)3 
have M = 



(365.256) 2 (11.11)3(26 .9983)3* 

By logarithms 16.6888 log. 1.222410x2 . 2.444820 
23984 log. 4.380000x3 . 13 .140000 

Logarithm of the numerator, . . 15. 584820 



MASSES OF THE PLANETS. 179 

365.256 log. 2.562600x2 . 5.125200 chap.hi. 

11.11 log. 1 .045714x3 . 3 .137142 

26.9983 log. 1 .431320x3 . 4 .293960 
Logarithm of the denominator, . . 12 .556302 
Therefore M= 1068* log. ... 3 .028518 
This result shows that the mass of the sun is 1068 times 
the mass of Jupiter ; but we previously found the mass of 
the sun to be 354945 times the mass of the earth, and if 
unity is taken for the mass of the earth, and J for the mass 
of Jupiter, we shall have 

1068 </= 354945; 

because each member of this equation is equal to the mass 
of the sun. 

By dividing both members of this equation by 1068, we The mass of 
find the mass of Jupiter to be 332 times that of the earth ; "redtoThat 
but in Art. 132, we found the bulk of Jupiter to be 1260 of the earth, 
times the bulk of the earth ; therefore the density of Jupiter 
is much less than the density of the earth. 

In the same manner we may find the masses of Saturn and The masses 
Uranus — the former is 105.6 times, and the latter 18.2 of Satnrn 
times the mass of the earth. 

The principles embraced in equation ( A ) apply only to 
those planets that have satellites ; for it is by the rapid or 
slow motion of such satellites that we determine the amount 
of the attractive force of the planet. 

In short, the masses of those planets which have satellites, what re- 
are known to great accuracy ; but the results attached to SBlts ma y be 

considered 

others in table IV, must be regarded as near approximations. acC urate. 

The slight variations which the earth's motion experiences The masses 
by the attractions of Yenus and Mars, are sufficiently sensi- Mars an( | 
ble to make known the masses of these planets ; and M. Mercury. 
Burckhardt gives ¥ oiVtt f° r Yenus, and 25 4 6 3 20 f° r Mars 
( the mass of the sun being unity ) ; Mercury he put down at 

* This is a correct result according to these data ; but more modern 
observations, in relation to the micromatic measure of Jupiter, and 
the distance of his satellites, give results a little different, as expressed 
in table IV. 



180 ASTRONOMY. 

Chap, in ---J-—- ; but this result is little more than hypothetical, 
as it is drawn from its volume, on the supposition that the 
densities of the planets are reciprocal to their mean distances 
from the sun ; which is nearly true for Venus, the earth, and 
Mars. 

J5 y means of / yi\. ) It may be astonishing, but it is nevertheless true, 

gravity and ^ ' J . , „ 

the lunar par- that by means of equations (-4) and (i>) we can find the 
aiiax, we diameter of the earth to a greater degree of exactness than by 

may find the 

diameter of an y one actual measurement. 

the earth. "We have several times observed that equation (A) is not 
accurate when used to find the masses of the earth and sun, 
because it contained the time of the revolution of the moon; 
which revolution is accelerated by the gravity of the moon, and 
retarded by the action of the sun. 

Therefore, to make equation (^4) accurately express the 
mass of the sun, the element t 2 requires two corrections, 
which will be determined by subsequent investigation. The 
first is an increase of T T ^ th part ; the second is a diminution 
of a^-gth part, and both corrections will be made if we take 
76-358 . , 
75^69** m Pkce ° f tK 
a common Then having two correct expressions for the mass of the 
sun, those two expressions must equal each other ; that is, 
76-358 t 2 D* ^ 2 D 3 
75-359 T 2 R* = 2gr*T 2 ' 

By suppressing common factors, we have 

76-358 * 2 7t 2 

75-359 R* ~~ %&&' 

In this equation r represents the mean radius of the earth, 
and we will suppose it unknown ; the equation will then 
make it known. 

The relation between R, the mean radius of the lunar or- 
bit, and r, the mean radius of the earth, is given by means 
of the moon's horizontal parallax. 
Equatorial The moon's equatorial horizontal parallax, as we have seen, 

horizontal ' J- ■ * '••*■* L 

parallax and (65) is 57' 3"; but the horizontal parallax for the mean ra- 



MASSES OF THE PLANETS. XgX 

dius, is 56^ 57"; this makes R = ( 60.36 ) r, whatever the chap. m. 
numerical value of r may be. Put this value of R in the T 

■* mean hon- 

preceding equation, and suppress the common factor r 2 , zontai parai- 

76-358 * 2 *•* 



we then have 



Therefore, 



75-359 ( 60.36 )*r 2g 

2<?-76-358* 2 



75-359(60.36)3^ 

As g is expressed in feet, and corresponds to t in seconds, confidence 
the numerical value of r will be in feet, which divided by inthe result - 
5280, the number of feet in a mile, will give the number of 
miles in the mean radius or mean semidiameter of the earth; 
and by applying the preceding equation, giving g, t, and &, their 
proper values ; and by the help of logarithms, we readily find 
r = 3953 miles ; only three miles from the most approved 
result; and we do not hesitate to say, that this result is more 
to be relied upon than any other. 

MASS OF THE MOON. 

( 172. ) Approximations to the mass of the moon have The mass of 
been determined, from time to time, by careful observations the moon 
on the tides ; but it is in vain to look for mathematical re- determined 
suits from this source ; for it is impossible to decide whether from obser- 
any particular tide has been accelerated or retarded, aug- v ^ tl0n . s on 
mented or diminished, by the winds and weather; and if not 
affected at the place of observation, it might have been at 
remote distances ; but notwithstanding this objection, the 
mass of the moon can be pretty accurately determined by 
means of the tides, owing to the great number and variety 
of observations that can be brought into the account; and 
we shall give an exposition of this deduction hereafter ; but 
at present we shall confine our attention to the following 
simple and elegant -method of obtaining the same result. 

If the moon had no mass ; that is, if it were a mere mate- 
rial point, and was not disturbed by the attraction of the 
sun, then the distance that the moon would fall from a tan- 
gent of its orbit, in one second of time, would be just equal 

p 



182 ASTRONOMY 

Chap. III. gr 2 

to — . (Art. 155. ) In this expression g, r, and R, repre- 

JTL 

sent the same quantities as in the last article. The dis- 
tance that the moon actually falls from a tangent of its orbit, 
in one second of time, is equal to the versed sine of the arc it 
describes in that time, and the analytical expression for it is 
found thus : 

Let «■ R represent the circumference of the lunar orbit, and if 
t is put for the number of seconds in a mean revolution, then 

nR 

— represents the arc corresponding to the moon's motion in 

one second (Fig. 30), and as this so nearly coincides with 
a chord, we have 

nR vR 7r*R 



7T 2 R 

Anexpres- Hence, we perceive, that -^-^- is the distance that the 

sion for the 



2t 2 



distance the moon wou \^ f a ]] f rom the tangent of its orbit in one second 

moon falls in ° 

one second of time, if it were undisturbed by the action of the sun ; but 

of time. g^g 

we can free it from such action by multiplying it by — ^, 

as we shall show in a subsequent chapter. That is, the 
attraction of both the earth and moon, at the distance of the 

■' . 859»* R 
lunar orbit, is g^g^i • 

But the attraction of' the earth alone, at the same distance, 

g t 2 
is -=- ; and comparing these quantities with the more gene- 

-LtJ 

ral expressions in Art. 156, we have 

E E+m gr 2 359 re *R 

R 2 '' R 2 :: 1& '' 358-2 W 

By suppressing the common denominator, in the first 
couplet, and calling E, the mass of the earth, unity, the pro- 
portion reduces to 

, , , 359^-^3 

1 : 1+m :: gT > : _^__ . 



MASSES OF THE PLANETS. 183 

As in the last article, i2=(60.36)r, and this value put for Chap - In - 
JR 3 , and reduced, gives 

359*-2(60.36) 3 r 
1 : 1-f-m : : g : ttkftt, — - — \ 

™ , -, ■ 359*r 2 (60.36) 3 r 

Therefore, - - l-\-m= 358-2 a 2 ? ' The result 

This fraction, as well as the one in the last article, can be 
reduced arithmetically; but the operation would be too 
tedious; they are both readily reduced by logarithms, by 
which we found l-L-wi=l. 01301 ; hence m=.01301, which 

Result 

is a little less than 7 y th. Laplace says -J^-th of the earth given by La- 
is the true mass of the moon ; and this value we shall use. P lace - 

THE DENSITIES OF BODIES. 

(173.) The density of a body is only a comparative term, standard 
and to find the comparison, some one body must be taken as for densit y- 
the standard of measure. The earth is generally taken for 
that standard. 

It is an axiom, in philosophy, that the same mass, in a 
smaller volume, must be greater in density; and larger in 
volume, must be less in density ; and, in short, the density 
must be directly proportional to the mass, and inversely pro- 
portional to the volume ; and if the earth is taken for unity 
in mass, and unity in volume, then it will be unity in density 
also ; and the density of any other planetary body will be its 
mass divided by its volume ; and if its volume is not given, the 
density may be found by the following proportion, in which 
d represents the density sought, and r the radii of the body ; 
the radius of the earth being unity. The proportion is drawn 
from the consideration that spheres are to one another as the 
cubes of their radii. 

1 mass ^ ■■ ■ , mass 

t : — — : : 1 : d; hence d = — — . 

From this equation we readily find the density of the sun, 
for we have its mass (354945), and its semidiameter 111.6 for tiTTe™ 
times the semidiameter of the earth (Art. 156) ; therefore its sities of 
13 



184 ASTRONOMY. 

Chap. III. _ . » 354945 '.,/, ... 

density must (be — =0.254, or a little more than ith 

spheres com- (111. O) 3 

pared to the ^ density of the earth. 

density of 

the earth. The mass of Jupiter is 332 times that of the earth, and its 

volume is 1260 times the volume of the earth ; therefore the 

332 
density of Jupiter is —0.264 ; which is a little more 

than the density of the sun. 
Densities The mass of the moon is -y 1 ^, and its volume J ¥ , therefore its 
moon &C. 61 ' density is T ^ divided by ^\, or ff =0.6533; about f the den- 
sity of the earth. 

From these examples the reader will understand how the 
densities were found, as expressed in table IY. 

GRAVITY ON THE SURFACE OF SPHERES. 

Gravity on ( 174.) The gravity on the surface of a sphere depends on 

of the other * ne mass an< ^ volume. The attraction on the surface of a 

planets, how sphere is the same as if its whole mass were collected at its 

center; and the greater the distance from the center to the 

surface, the less the attraction, in proportion to the square of 

the distance : but here, as in the last article, some one sphere 

must be taken for the unit, and we take the earth, as before. 

The mass of the sun is 354945, and the distance from its 

center to its surface is 111.6 times the semidiameter of the 

earth ; therefore a pound, on the surface of the earth, is to 

the pressure of the same mass, if it were on the surface of 

1 354945 , 

the sun, as - to — — — - — , or as 1 to 2b nearly. That 
1 (111.6) 2 J 

is, one pound on the surface of the earth would be nearly 28 

pounds on the surface of the sun, if transported thither. 

The mass of Jupiter is 332, and its radius, compared to 

that of the earth, is 11.1 (Art. 131) ; therefore one pound, on 

332 

the surface of the earth, would be , or 2.48 pounds on 

the surface of Jupiter; and by the same principle, we can 
compute the pressure on the surface of any other planet. 
Results will be found in table IV. 



LUNAR PERTURBATIONS. 185 



CHAPTER IV. 

PROBLEM OF THE THREE BODIES. LUNAR PERTURBATIONS. 

(175.) By the theory of universal gravitation, everybody in Chap. iv. 
the universe attracts every other body, in proportion to its The theory 
mass; and inversely as the square of its distance; but simple ° gnm} " 
and unexceptionable as the law really is, it produces very com- 
plicated results, in the motions of the heavenly bodies. 

If there were but two bodies in the universe, their motions The com- 
would be comparatively simple, and easily traced, for they p exl * y 
would either fall together or circulate around each other in 
some one undeviating curve; but as it is, when two bodies 
circulate around each other, every other body causes a devia- 
tion or vibration from that primary curve that they would 
otherwise have. 

The final result of a multitude of conflicting motions can- 
not be ascertained by considering the whole in mass ; we must 
take the disturbance of one body at a time, and settle upon 
its results ; then another and another, and so on ; and the sum 
of the results will be the final result sought. 

We, then, consider two bodies in motion disturbed by a The prob- 
third bodv ; and to find all its results, in general terms, is ? m the 

J . three bodies. 

the famous problem of "the three bodies;" but its complete 
solution surpasses the power of analysis, and the most skillful 
mathematician is obliged to content himself with approxi- 
mations and special cases. Happily, however, the masses of 
most of the planets are so small, in comparison with the mass 
of the sun, and their distances so great, that their influences 
are insensible. 

We shall make no attempt to give minute results ; but we 
hope to show general principles in such a manner, that the 
reader may comprehend the common inequalities of planetary 
motions. 

Let m, Fig. 34, be the position of a body circulating around Abstract 
another body, A, moving in the direction PmB, and dis- attractl0n - 
turbed by the attraction of some distant body, D. 



186 



ASTRONOMY. 



Chap TV. 



Two bodies 
equally at- 
tracted in pa- 
rallel lines 
are not af- 
fected in 
their mutual 
relations. 




We now propose to show some of 
the most general effects of the ac- 
tion of D, luithout paying the least re~ 
gard to quantity. 

If A and m were equally at- 
tracted by D, and the attraction 
exerted in parallel lines, then D would 
not disturb the mutual relations of 
A and m. But while m is nearer to 
D than A is to D, it must be more 
strongly attracted, and let the line 
mp represent this excess of attraction. 
Decompose this force (see Nat. Phil.) 
into two others, mn and np, the first 
along the line A m, the other at right 
angles to it. 

The first is a lifting force ( called 
by astronomers the radial force), 
the other is a tang ental force, and affects the motion of m. It 
will accelerate the motion of m, while acting with it, from P 
to JB, and retard its motion, while acting against it, from B 
to Q. 

We must now examine the effect, when the revolving body 
is at m', a greater distance from D than A is from D. 

Now A is more strongly attracted than m'\ and the result 
of this unequal attraction is the same as though A were not 
attracted at all, and m! attracted the other way by a force 
equal to the difference of the attractions of D on the two 
bodies A and m' . Let this difference be represented by the 
line w! p', and decompose it into two other forces, m! n' and 
n'p', the first a lifting force, the other the tangental force. 

The rationale of this last position may not be perceived by 
every reader, and to such we suggest, that they conceive A 
and m! joined together by an inflexible line, A m', and both 
A and m' drawn toward D, but A drawn a greater dis- 
tance than m'. Then it is plain that the position of the line 
A w! will be changed ; the angle D A m' will become greater, 
and the gngle CAm' less — that is, the motion of m! will be 



LUNAR PERTURBATIONS, 



187 



accelerated from Q to <X but from C to P it will be re- Chap, iv . 
tarded. 

In short, the motion of m will he accelerated when moving to- The dis- 
ward the line BBC, and retarded while moving from that line. ^"sjj * 7 
That is, retarded from B to Q, accelerated from Q to C, re- urgeS arevoi- 
tarded from C to P, and again accelerated from P to B. ving body to 

If we conceive A to be the earth, m the moon, and D the * * ™ 
sun; then D B C is called the line of the syzigies, a term 
which means the plane in which conjunctions and oppositions 
take place. At the point B the moon falls in conjunction with 
the sun, and is new moon ; at the point C it is in opposition, 



Fig. 



36. 

D 




Action of 
an attracting 
body on a 
ring. 



or full moon. 

( 176. ) Conceive a ring of matter around 
a sphere, as represented in Fig. 36, and let 
it be either attached or detached from the 
sphere, and let D be not in the plane of the 
ring. 

From what was explained in the last ar- 
ticle, the particles of matter at m are con- 
stantly urged toward the line D B C, and 
the particles at m' are constantly urged 
toward the same plane; that is, the at- 
traction of D, on the ring, has a tendency to 
diminish its inclination to the line D B C; 
and its position would be changed by such 
attraction from what it would otherwise be ; 
and if the ring is attached to the sphere, the sphere itself will 
have a slight motion in consequence of the action on the ring. 

Now there is, in fact, a broad ring attached to the equator- 
ial part of the earth, giving the whole a spheroidal form ; and 
the plane of the equator is in the plane of the ring. 

When the sun or moon is without the plane of this ring, cause 
that is, without the plane of the equator, their attraction has notation 
a tendency to draw the plane of the equator toward the at- 
tracting body, and actually does so draw it ; which motion is 
called nutation. How this motion was discovered, and its 
amount ascertained, will be explained in a subsequent chapter. 
(177.) We may conceive the line DBC to be in the 



7Tb 



188 



ASTRONOMY. 



Chap. IV. 

Applica- 
tion of the 
ring to the 
lunar orbit. 



The moon'; 
nodes retro' 
srrade. 



Lnnar per 
turbations. 



Investiga- 
tion for find- 
ing a general 
analytical 
expression 
for the lunar 
perturba- 
tions. 



plane of the ecliptic, D the sun, and the ring around the earth 
the moon's orbit, inclined to the plane of the ecliptic with an 
angle of about Jive degrees ; then when the sun is out of the 
plane of the ring, or moon's orbit, the action of the sun has 
a constant tendency to bring the moon into the ecliptic, and 
by this tendency the moon does fall into the ecliptic from 
either side sooner than it otherwise would. 

The point where the moon falls into the ecliptic is called 
the mooris node; and by this external action of the sun the 

moon falls into the ecliptic 
from its greatest inclination 
before it describes 90°, and 
goes from node to node be- 
fore it describes 180° — and 
hence we say that the moon's 
nodes fall backward on the 
ecliptic. The rate of retro- 
gradation is 19° 19' in a year, 
making a whole circle in about 
18.6 years. 

( 178. ) We are now pre- 
pared to be a little more defi- 
nite, and inquire as to the 
amount of some of the lunar 
irregularities. 

Let S be the mass of the 
sun, E that of the earth, and 
m the moon, situated at D. 
Let a be the mean distance 
between the earth and sun, z 
the distance between the sun 
and moon, and r the mean ra- 
dius of the lunar orbit. Let 
the moon have any indefinite 
position in its orbit. ( It is 




represented in the figure at D. ) 



S 



The attraction of the sun on the earth is — -, the attrac- 
ts 2 



LUNAR PERTURBATIONS 189 



tion of the sun on the moon is — ; and the attraction of the - — ' 

earth and moon, on the moon, is *"" . ( Art. 156. ) 

Let the line D B, the diagonal of the parallelogram A C, be 
the attraction of the sun on the moon, and decompose it into 
the two forces D A and D 0; the first along the lunar radius 
vector, the other parallel to SE. 

The two triangles C D B and D S E are similar, and give 

S 
the proportion a : z : : CB : B B. But BB= — ; 

a S' 
Therefore CD = . By a similar proportion we find 

D A = . 

Let the angle S E D be represented by x, then D G will 
be expressed by r cos. x, and SB will be a right line nearly, 
for the angle B S E is never greater than 1'. 

Now if the force B C, which is parallel to SE, is only 
equal to the force of the sun's attraction on the earth, it 
will not disturb the mutual relations of the earth and moon. 

The force of the sun's attraction on the earth is — ; and as this 

or 

must be less than the force of attraction on the moon, when 
the moon is at B, conceive it represented by the line Cn, and 
subtracted from CD, will leave Bn the excess of the sun's 
attraction on the two bodies, the earth and the moon ; and 
this alone constitutes the disturbing force of the moon's 
motion ; 

That is, Bn = CB—Cn = ?f— 4 ; An expres " 

Z 3 a 2 sion for the 



Or Dn = aS ( — — ), the disturbing force. Decom- 
pose this force ( Dn ) into two others, Dp and pn, by means 
of the right angled triangle Dpn; the angle pDn being 
equal to DE S, which we represent by x. 



whole distur- 
bing force. 



190 ASTRONOMY. 

Chap. IV . / 1 1 \ 

Whence Dp = Sa ( ~ — — ) cob. x; 

And pn = Sa ( — ) sin. x. 

\z 3 a 3 / 

/ r£\ 

The force DA, i.e. ( — ) is called the additions force; 

The radial the force Dp the ablatitious force. The difference of these 
forc€ - two forces is called the radial force ; that is 



Sa 



( 1 1\ rS 

I J cos. x = the radial force ; pn is the 

\z 3 a 3 / z 3 ' r 

tangental force. 

Expression when the angle x is equal to 90°, cos. x = o, S D — SE, 

of the rarlial 

orce at t« on=a . which values, substituted, give for the value 

quadratures. ' ' ° ^3 

of the radial force at the quadratures, and its tendency there 
is to increase the gravity of the moon to the earth. When 
the angle x is zero ( the moon is in conjunction with the sun ) 
the cos. x = 1, and the radial force becomes 

Sa Sa rS S ( a — r ) Sa 

• or — i . 

z 3 a 3 z 3 z 3 a 3 

But at that point z = (a — r ), which value substituted, 

and rejecting the comparatively very small quantities in both 

numerator and denominator, we have, for the radial force at 

2rS 

conjunction, . 

a 3 

When the angle x = 180° ( the moon is in opposition to 

the sun ), cos. x = — 1, and the force becomes 

Sa Sa rS S S(a4-r) 

• or ^ ! — ' m 

a 3 z 3 z 3 a 2 z 3 

But at this point z = a -(- r, which, substituting as before, 

2rS 

and we have for the radial force in opposition , the same 

a 3 

expression as at conjunction. 

If we compare the radial force at the syzigies with the ex- 
pression for it at the quadratures, we shall find it the same 
in form, but double in amount and opposite in sign, showing 
that it is opposite in effect. 



LUNAR PERTURBATIONS. 191 

( 179. ) As the radial force increases the gravity of the Chap. iv. 
moon to the earth, at the quadratures, and diminishes it at 
the syzigies, there must be points in the orbit symmetrically W h ere thera- 
situated, in respect to the syzigies, where the radial force dial force is 
neither increases nor diminishes the gravity, and of course zer °" 
its expression for those points must be zero; and to find How to 
these points we must have the equation find them - 



Sa 



{ ) cos. x r=0 . . (1) 

\z 3 a 3 / z 3 y J 



By inspecting the figure we perceive that the line SD G 
is in value nearly equal to the line S E, and for all points in 
the orbit we have 

z = a + r cos. x ( 2 ) 

Reducing equation ( 1 ) we have 

(a 3 — z 3 ) cos. xz=ra 2 . . . . (3) 
Cubing ( 2 ) 

z 3 = a 3 -j^Ba 2 r cos. x j^L Bar 2 cos. 2 x + r 3 cos. 3 x. 
As r is very small in relation to a, the terms containing the 
powers of r, after the first, may be rejected; we then have 

(a 3 — z 3 ) = + 3a 2 r cos. x. . . (4) 
This value substituted in ( 3 ), and reduced, gives 

— Q o -J ReSUlt 0f 

4- 3 C0S - ** = *' the radial 

Hence cos. x = J\ and x = 54° 44', or the points force at the 

~ 6 x quadratures 

are 35° 16' from the quadratures. *nd syzigies, 

This shows that at the quadratures, and about 35° on 
each side of them, the gravity of the moon is increased by 
the action of the sun, and at the syzigies, and about 54° on 
each side of them, the gravity is diminished ; and the diminu- 
tion in the one case is double the amount of increase in the Mean ra- 
other, and by the application of the differential calculus we 
learn that the mean result, for the entire revolution, is a dimi- 

nution whose analytical expression is ^-^ ; an expression 

which holds a very prominent place in the lunar theory; the 



192 ASTRONOMY. 

Chap - sv - result of which we have used in Art. 171, and there stated it 
to be 3- jg-th part of the force that retained the moon in its 
orbit. 
Value of But how do we know this to be its numerical value, is a 
jie mean ra- y serious inquiry of the critical student? 

ilial force, " x J 

, , ft] I rjm 

and now The force that retains the moon in its orbit is - 

found. 



( Art. 156 ) ; and if the radial force can be rendered homoge- 
neous with this, some numerical ratio must exist between 
them. Let x represent that ratio, and we must find some 
numerical value for x to satisfy the following equation : 
rS _ E-\-m 

Tner ef 0re x= HI+pl 3 ; 

calling E=l, m == T V (Art. 172), or E -\- m is 1.013. 
S = 354945 ( Art. 169 ), and the relation between the 
mean distance to the sun, and the mean radius of the lunar 
orbit, is 397.3,* therefore 

(2.026)(397.3)3 

* 154945 Sm ' 

or the coefficient to x, in equation ( A ), is one three hundredth 
and fifty -eighth part of the force which retains the moon in its 
orbit. 
General ef- (180.) The mean radial force causes the moon to circu- 
aiai force * ate at ik^ P ar<; greater distance from the earth than it 
otherwise would have, and its periodical revolution is in- 
creased by its 179th part ; but this would cause no variation 
or irregularity in its distance or angular motion, provided its 
orbit were circular, and the earth and moon always at the 
same mean distance from the sun. 

rS 
The radial But we perceive the expression ~-j contains two variable 

force varia- 

ble - quantities, r and a, which are not always the same in value ; 

and, therefore, the value of the expression itself must be va- 

* This relation is found by dividing the horizontal parallax of the 
moon, 56' 57", by the horizontal parallax of the sun, 8".6. 



LUNAR PERTURBATIONS. 193 

riable ; and it will be least when the earth is at the greatest Chap. tv. 

distance from the sun, and, of course, the moon's motion will 

then be increased. But the earth's variable distance from 

the sun depends on the eccentricity of the earth's orbit ; and The annu* 

hence we perceive that the same cause which affects the ap- a e( i uatl0n 

x *- of the moons 

parent solar motion, affects also the motion of the moon, and motion, 
gives rise to an equation called the annual equation* of the 
moon's motion. It amounts to 11' in its maximum, and va- 
ries by the same law as the equation of the sun's center. 

(181.) If we take the general expressions for the radial a general 

expression 
/. a ( -*- i *** i i • i xi. i j.j. f° r the radial 

torce, b a\ — — — I cos. x — — , and banish the letter z . 

\ z 3 a z/ ^ 3 ' force at any 

. . point of the 

trom it by means ot the equation moon's orbit 

z = a -j- r cos. x 

Or, 2 s = a 3 + 3a 2 r cos. x, 

(neglecting the powers of r) and we shall have, 

rS (3 cos. 2 x — 1) 
_ 

for an expression of the radial force corresponding to any 
angle x from the syzigy. 

If we take the general expression for the line pn, the tan- 
gental force, and banish z, as before, we have, 

. _ Srs cos. x sin. x 
tangental force, = . 



By doubling numerator and denominator this fraction can Expression 

for the tan- 
gental force. 



take the Mowing form : 



3rs (2 cos. x sin. x) 

But, by trigonometry, 2 cos. x sin. x = sin. 2x, 

Therefore the tangental force = — - — . 

6 2a 3 

This expression vanishes when x = o and x = 90° ; for then its vanish- 
sin. 2x = sin. 180 = 0. Hence the tangental force van- in ?P oints 
ishes at the syzigies and quadratures, attains its maximum 

* This is equation I, in the Lunar Tables. 
13 Q 



194 



ASTRONOMY. 



Chap. IV. 

The tan- 
gental force 
greatest 
when the 
earth is in 
perigee. 



Application 
of the radial 
force to an 
elliptical or- 
bit. 



value at the octants, and varies as the sine of the double angular 
distance of the moon from the sun. 

The mean maximum for this force must be determined by 
observation. It is known by the name of variation, and by 
mere inspection we can see that its amount must correspond 
to the variations of r and of a 3 . Hence, to obtain the moon's 
place, we must have correction on correction. 

The variation amounts to about 35'. It increases the ve- 
locity of the moon from the quadratures to the syzigies, and 
diminishes it from the syzigies to the quadratures ; hence, in 
consequence of the variation, the velocity of the moon is 
greatest at the syzigies, and least at the quadratures. 

( 182.) Let us now examine the effect of the radial force 
on the lunar orbit, considered as elliptical. 



#0 



Fig. 38. 



When the 
radial force 



Let SE (Fig. 37) be at right 
angles to A B, the greater axis 
of the lunar orbit, and conceive 
A C B to represent the orbit that 
the moon would take if it were 
undisturbed by the sun. 

But when the moon comes 

round to its perigee at A, it is in 

one of its quadratures, and the 

radial force then increases the 

gravity of the moon toward the 

rs 
earth by the expression — . But 

here r is less than its mean value, 
and the expression is less than its 
mean, and therefore the moon is 
j$ not crowded so near the earth as 
it otherwise would be, and, of 
course, at this point the moon 
D will run farther from the earth. 

At the point C, the radial force tends to increase the dis- 
tance between the earth and moon, and to widen the orbit. 

When the moon passes round to B, the radial force again 
increases the gravity of the moon, and r, in the expression 




LUNAR PERTURBATIONS. 



195 



rs 



, is greater than its mean value ; and, of course, crowds the 



Chap. IV. 



decreases 



so 



Fig. 39. 



When the 
radial force 
increases the 
eccentricity 
of the lunar 
orbit. 



moon nearer to the earth than it otherwise would go ; and the eccentri - 

• «pi T-if i city of the lu- 

thus we perceive that the action of the radial force on an el- nar ellipse, 
liptical orbit has a tendency to decrease the eccentricity of the 
ellipse, when the sun is at right angles to its greater axis. 

( 183.) Now conceive the sun to be in a line, or nearly in 
a line, with the longer axis of the lunar orbit, as represented 
in Fig. 38. 

The radial force at the quadratures, 
C and J), has a tendency to press in 
the orbit, or narrow it. At the point 
A, the tendency, it is true, is to in- 
crease the distance between the earth 
and moon ; but that tendency is not 
so strong as it would be if the moon 
were at its mean distance from the 
earth. 

The tendency at B is to increase 
the distance, and it is a tendency 
greater than the medium. That is, 
the tendency at A is less than the 
medium; at B, greater than the me- 
dium; and at C and D, the com- 
pressed parts of the orbit, the ten- 
dency is to a still greater compres- 
sion; therefore, the entire action of 
the radial force is to increase the ec- 
centricity of the lunar orbit, when the 
sun is in line, or nearly in line, with 
the longer axis. 

Thus, we perceive, that under the disturbing action of the 
sun, the eccentricity of the moon's orbit must be in a state 
of perpetual change, now more, now less, than its mean state. 

Corresponding with this change of eccentricity there must 
be changes in the lunar motion ; and to keep account of it, 
and allow for it, astronomers have formed a table called 
evection. 




196 



ASTRONOMY. 



Chap. IV. 

Effect of 
the radial 
force on the 
motion of'the 

lunar p.eri- 
.?ee. 



Fig. 40. 



O 
S 



O 

S' 



Retrograde 
motion of the 
perigee and 
apogee. 



The major 
axis of the 
lunar orbit is 
inclined to 
follow the 
sun. 



(184.) Now let us examine the effect of the radial force 
on the position of the lunar apogee. 

Let E (Fig. 40), he the earth, and, 
for the sake of simplicity, we conceive 
the earth to be stationary, and the 
sun and moon both to revolve about 
it with their apparent angular veloci- 
ties ; the moon in the orbit A C B, 
and in the direction A C B; the 
sun in a distant orbit, part of which 
is represented by S S'. 

Let A B be the greater axis of the 
moon's orbit, in its natural position, 
or as it would be if undisturbed by 
the sun; and being undisturbed, the 
perigee and apogee would remain con- 
stant at the points A and B, and the 
time from A to B, or from B to A, 
would be just equal to the mean time 
of half a revolution, as explained in 
a former part of this work. 

Now let us conceive the sun to be 
in its orbit at S, then the moon will 
be in the syzigy when it comes round 
to s, and as the radial force at that point tends to increase 
the distance between the earth and the moon, the apogee will 
take place at s, or between s and B; and it is evident that 
the apogee in that case would recede or run back. But at 
the next revolution of the moon, in a little more than twenty- 
seven days, the sun at that time will, apparently, have moved 
to S' about twenty-seven degrees. Now the syzigy will take 
place at s ', and the greatest distance between the earth and 
moon will now be between B and s\ that is, the apogee will 
advance, in one revolution, from near s to near s f ; and thus, 
in general, the longer axis of the moon's orbit is strongly in- 
clined to follow the sun ; and this is the source of its pro- 
gressive motion. It makes a revolution in 3232i days ; 
but its motion is very irregular, for, as we have just seen, 




LUNAR PERTURBATIONS. 197 

when the line which joins the earth and sun makes a very chap. iv. 
acute angle with the longer axis of the lunar orbit, and is ap- 
proaching that axis, the motion of the apogee and perigee is 
retrograde; but, all of a sudden, when the sun passes the 
longer axis of the lunar orbit, the motion of the apogee be- 
comes direct, and moves with considerable rapidity. 

When the sun is at right angles to the major axis of the Under what 
moon's orbit, the tendency of the radial force is to diminish P osition of 

. . -. , tne sun the 

the eccentricity of the orbit, but it has no tendency to change lunar perigee 

the position Of the axis. remains sta- 

From this investigation it follows, that when the sun has 
just passed the greater axis of the lunar orbit, the interval 
from apogee to apogee, or from perigee to perigee, will be 
greater than a revolution. Just before the sun arrives at the 
position of the longer axis, the time from one apogee to an- 
other is less than a revolution; and when the sun is at 
right angles to the longer axis, the time is just equal to a 
revolution in longitude. 

(185.) By comparing eclipses of the moon, observed by Ancient 
the ancient Egyptians and Chaldeans, with those of more ecipsei 

®T r pared with 

modern times, Dr. Halley, and other astronomers, concluded mo dem ob- 
that the periodic time of the moon is now a little shorter stations. 
than at those remote periods; and to make these extreme 
observations agree with modern ones, it became necessary 
to conceive the moon's mean motion to be accelerated about 
11 seconds per century. 

For a long time this fact seriously perplexed astronomers ; rrhs re- 
some were for condemning the theory of gravity as insuffi- 
cient to explain the cause of the lunar perturbations, while 
others were for rejecting the facts, although as well estab- 
lished as any mere historical facts could be. 

In this dilemma, says Herschel, "Laplace stepped in to 
rescue physical astronomy from reproach by pointing out the 
real cause of the phenomenon in question." 

Although this subject troubled the greatest philosophers 
of the past age — the greatest mathematical philosophers the 
world ever saw — the problem is quite simple, now the solu- 
tion is pointed out, and we are sure that every reader of or- 



198 ASTRONOMY. 

Chap, iv . dinary capacity can understand it, provided he gives his se- 
rious attention to the subject. 
a summary rj^g gecu i ar acceleration of the moon's mean motion is 

statement of 

the cause, caused by a small change in the mean value of the radial force, 

occasioned by a change in the eccentricity of the earth's orbit. 

rS 
The expression - — is the mean radial force of the sun 

acting on the moon's orbit, dilating it and increasing the 
time of the lunar revolution. 
when the jf ^ e ear th's orbit had no eccentricity, 2a 3 , the denomina- 
tion is in- * or °f tne fraction, would always have the same value, and 
creased. then regarding the numerator as constant, there would be no 
variation of the moon's motion arising from this cause. But 
in consequence of the earth and moon moving toward the 
apogee of the earth's orbit, a, of course, a 3 becomes 
greater, and the value of the radial force becomes less than 
its mean value, and in consequence of this, the moon's mo- 
tion is increased. And when the earth and moon move to- 
when di- ward the earth's perigee, a and a 3 become less, and the 
value of the radial force becomes greater than its mean ; the 
moon's orbit is dilated to excess, and its motion is diminished ; 
The ex- and the orbit is more dilated when the earth is in perigee than it 
pression or ^ contracted when the earth is in apogee. In other words, the 

the mean ra- . 

dial force is mean dilatation of the lunar orbit is greater, and the mean 
not the true mo tion of the moon less, in proportion as the earth's orbit is 

mean. . . 

more eccentric. 

rS 
The less the value of ~ — the greater is the moon's mean 

motion, and that value is least when a is greatest. But a 

would have no variation of value if the earth's orbit were 

circular. 

The earth's orbit, however, is eccentric, and in the course 

of a year the value of the radial force is exactly expressed 

rS 
by q — only at two instants of time, when the earth passes 

the extremities of the shorter axis of its orbit. At all 
other times a is either greater or less than its mean 
value, and the variations are equal on each side of it ; that 



LUNAR PERTURBATIONS. 199 

is, a becomes (a — d) or (a-{-d), and the radial force is Chap. iv. 

really 

rS rS 

-— or 



2(a—d) 3 2(a+d) 3 ' 

which expressions correspond to equal distances on each side The true 
of the mean distance, and d may have all values, from to of the radial 
a e, the eccentricity. The mean value of the radial force *>«». 
corresponding to the whole year, is equal to 

V jrS ±rS \ 

2\(o— d)*^ {a+dy/' 



Q rS/ 1 1 \ 

ur ' 4 \( a —d)3~T~(a4-d) 3 ' 



(a—dy ■ (a+d)' 

rS 

But this expression is always greater than - — , except The mean 

La 3 value of the 

when d — ; then it is the same, as any algebraist can verify. ra< Jj a ^ f 106 

Will D6 l63.St 

Hence the mean radial force for the whole year is greater f ail when 

as the earth's orbit is more eccentric, and it will be least of the . earth ' s 

all when that orbit becomes a circle ; and then, and then circ i e> 

rS 
only, it will be accurately represented by ^--. 

But when the radial force is least, the mean motion must 
be greatest, and that force is less and less as the eccentricity 
of the earth's orbit becomes less and less; and corresponding 
thereto the moon's motion becomes greater and greater, as 
has been the case for more than 4000 years. 

( 186. ) The mean distance between the earth and sun re- The cause 
mains constant. It must be so from the nature of motion, ° f e e ^^ m 
force, action, and reaction ; but by the attraction of the city of the 
planets the eccentricity of the earth's orbit is in a state of per- earth ' s orblt ' 
petual change ; the change, however, is excessively slow. From 
the earliest ages the eccentricity of the orbit has been dimin- 
ishing ; and this diminution will probably continue until it is 
annihilated altogether, and the orbit becomes a circle; after 
which it will open out in another direction, again become ec- 
centric, and increase in eccentricity to a certain moderate 
amount, and then again decrease. 
14 



200 ASTRONOMY. 

chap, iv . The period for these vibrations, " though calculable, has never 

The im- oeen calculated further than to satisfy us that it is not to be 

cones 8 on- reckoned by hundreds or even by thousands of years." It is a 

ding to these period so long that the history of astronomy, and of the whole 

changes. human race, is but a point in comparison. 

The moon's mean motion will continue to increase until the 

earth's orbit becomes a circle; after which it will again decrease, 

corresponding with the increase of a new eccentricity. 

The mch- (187.) For the sake of simplicity, we have thus far con- 
nation of the v y . . , 
innar orbit sidered the moon s orbit to be in the same plane as the 

taken into ear th's orbit ; but this is not true; the mean inclination of the 
lunar orbit to the ecliptic is 5° 8', varying about 9' each way, 
according to the position of the sun. 

Owing to this inclination of the lunar orbit, the expressions 
which we have obtained for the tangental force need cor- 
rection, by multiplying them by the cosine of the inclination ; 
and for the effect of the same forces in a perpendicular 
direction to the moon's longitude, multiply them by the sine 
of the inclination of the orbit. 

The position of the moon's orbit, in relation to the sun, is 
strictly analogous to the ring in relation to the disturbing 
body D (Art. 176) ; the sun is constantly urging the moon 
into the plane of the ecliptic, which has a constant tendency 
to diminish the inclination of the lunar orbit ( except when 
the sun is in the positions of the moon's nodes) ; and this con- 
stant force urging the moon to the ecliptic, causes the moon's 
nodes to retrograde. 

We conclude this chapter by a brief summary of the prin- 
cipal causes which affect the moon's motion. 
a summary 1. The eccentricity of the earth's orbit ; which gives rise to 
S^iuhaf il tne annual equation of the moon in longitude. 
regularities. 2. The eccentricity of the lunar orbit ; producing the equa- 
tion of the center. 

3. The tangental force; giving rise to the equation called 
variation. 

4. The position of the sun in respect to the greater axis 
of the lunar orbit; giving rise to the inequality called evection. 

5. The inclination of the moon's orbit. 



THE TIDES. 201 

6. The combination of the first cause, when differing from chap. iv. 
its mean state, augments or diminishes the result of every 

other — thus making many additional small equations. 

7. The ellipsoidal form of the earth. 



CHAPTER V. 

THE TIDES. 

( 188. ) The alternate rise and fall of the surface of the chap. v. 
sea, as observed at all places directly connected with the Definition 
waters of the ocean, is called tide ; and before its cause was of the term 

tide. 

definitely known, it was recognized as having some hidden and 
mysterious connection with the moon, for it rose and fell twice Connection 
in every lunar dav- High water and low water had no con- W1 

^ jo moon. 

nection with the hour of the day, but it always occurred in 
about suck an interval of time after the moon had passed the 
meridian. 

When the sun and moon were in conjunction, or in opposi- High tides, 
tion, the tides were observed to be higher than usual. 

When the moon was nearest the earth, in her perigee, other 
circumstances being equal, the tides were observed to be 
higher than when, under the same circumstances, the moon 
was in her apogee. 

The space of time from one tide to another, or from 
high water to high water ( when undisturbed by wind ), is 
12 hours and about 24 minutes, thus making two tides in one 
lunar da}^ ; showing high water on opposite sides of the earth 
at the same time. 

The declination of the moon, also, has a very sensible influ- Tides af- 
ence on the tides. When the declination is high in the north, f c1 * y * e 

° declination 

the tide in the northern hemisphere, which is next to the moon, f the moon. 
is greater than the opposite tide ; and when the declination of 
the moon is south, the tide opposite to the moon is greatest. A difficulty 
It is considered mysterious, by most persons, that the moon a snperficial 
by its attraction should be able to raise a tide on the opposite reasoner. 
side of the earth. 



202 



ASTRONOMY. 



The true 
cause. 



Fig. 41. 
m 



chap. v. That the moon should attract the water on the side of the 
earth next to her, and thereby raise a tide, seems rational and 
natural, but that the same simple action also raises the oppo- 
site tide, is not readily admitted ; and, in the absence of clear 
illustration, it has often excited mental rebellion — and not a 
few popular lecturers have attempted explanations from false 
and inadequate causes. 

But the true cause is the sun and moon's attraction; and 

until this is clearly and decidedly 
understood — not merely assented 
to, but fully comprehended — it is 
impossible to understand the com- 
mon results of the theory of gra- 
vity, which are constantly exem- 
plified in the solar system. 

We now give a rude, but strik- 
ing, and, we hope, a satisfactory 
explanation. 

Conceive the frame- work of the 
earth to be an inflexible solid, as it 
really is, composed of rock, and in- 
capable of changing its form under 
any degree of attraction ; conceive 
also that this solid protuberates 
out of the sea, at opposite points of 
the earth, at A and B, as repre- 
sented in Fig. 41, A being on the 
side of the earth next to the moon, 
m, and B opposite to it. Now in 
connection with this solid con- 
ceive a great portion of the earth 
to be composed of water, whose 
particles are inert, but readily 
move among themselves. 
The solid A B cannot expand under the moon's attrac- 
tion, and if it move, the whole mass moves together, in virtue 
of the moon's attraction, on its center of gravity. But the 
particles of water at a, being free to move, and being under a 



A summary 
illustration, 
of the tides. 




THE TIDES. 203 

more powerful attraction than the solid, rise toward A, pro- Chap. v. 
ducing a tide. 

The particles of water at b being less attracted toward m 
than the solid, will not move toward m as fast as the 
solid, and being inert, they will be, as it were, left behind. 
The solid is drawn toward the moon more powerfully than the 
particles of water at b, and sinks in part into the water, but 
the observer at B, of course, conceives it the water rising up 
on the shore (which in effect it is), thereby producing a 
tide. 

( 189. ) The mathematical astronomer perceives a strict Analogy 
analogy between the analytical expressions for the tides and lunar rtur _ 
the expressions for the perturbations of the lunar motion. bations and 

What we have called the radial force, in treating of the \ e per ! ur *" 

J ' c tions of the 

lunar irregularities, is the same in its nature as the force that ocean, 
raises the tides ; the tide force is a radial force, which dimi- 
nishes the pressure of the water toward the center of the 
earth under and opposite to the moon, in the same manner as 
the radial force diminishes the gravity of the moon toward 
the earth in her syzigies. 

In Art. 179 we found that the radial force for the moon, at The radial 

tr, o force as ap- 

the syzigies, is expressed by ■ ; in which expression S is P lied t0 the 

(X- moon. 

the mass of the sun, a its distance from the earth, and r the 
radius of the lunar orbit. 

The same expression is true for the tides, if we change S to Converted 

. into an ex. 

m, the mass of the moon, and conceive a to represent the dis- pre ssion for 
tance to the moon, and r the radius of the earth. For the the tides - 

tides, then, we have — — , and as the numerator is always con- 
stant, the variation of the tides must correspond to the cube 
of the inverse distance to the moon. 

( 190.) The sun's attraction on the earth is vastly greater Sun ' s at ' 

,. traction con* 

than that of the moon ; but by reason of the great distance S id er ed. 
to the sun, that body attracts every part of the earth nearly 
alike, and, therefore, it has much less influence in raising a 
tide than the moon. 



204 ASTRONOMY. 

Chap, v . From a long course of observations made at Brest, in 

observations France, it has been decided that the medium high tides, 

when the sun and moon act together in the syzigies, is 

19.317 feet; and when they act against each other (the 

moon in quadrature), the tides are only 9.151 feet. Hence 

compara- ^e efficacy of the moon, in producing the tides, is to that 

tive influen- 

cesofthesun °* "$ snn > as ™ e number 14.23 to 5.08. 

and moon. Among the islands in the Pacific ocean, observations give 
the proportion of 5 to 2.2, for the relative influences of these 
two bodies ; and, as this locality is more favorable to accu- 
racy than that of Brest, it is the proportion generally taken. 
Having the relative influences of two bodies in raising the 
tides, we have the relative masses of those two bodies, pro- 
vided they are at the same distance. But by the expression 
for the tides, as we have just seen, the variation for distance 
corresponds with the inverse cube of the distance, and the dis- 
tance to the sun is 397.2 times the mean distance to the 
moon. Hence, to have the influence of the moon on the 
tides, when that body is removed to the distance of the sun, 
we must divide its observed influence by the cube of 397.2. 
That is, the mass of the moon is, to the mass of the sun, as 

moon com- 
puted. 5 

the number .^tt— — -— to the number 2.2. 
(397.2) 3 

In all preceding computations we have called the mass of 
the earth unity, and in relation thereto, the mass of the sun is 
354945 (Art. 169). Let us represent the mass of the moon 
by m, then we have the following proportion : 

5 

The result. m : 354945 : : 7777^^— : 2.2. 

(397.2)3 

This proportion makes the mass of the moon a little less than 
-yL ; but I have little confidence in the accuracy of the result, 
as the data, from their very nature, must be vague and in- 
definite. 
The times (191.) The time of high water at any given point is not 
° iL Wa * commonly at the time the moon is on the meridian, but two 

ter different J ' 

in different or three hours after, owing to the inertia of the water ; and 
localities. p] acegj not far from each other, have high water at very dif- 



Mass of the 



THE TIDES. 205 

ferent times on the same day, according to the distance and Chap. v. 
direction that the tide wave has to undulate from the main 
ocean. 

The interval between the meridian passage of the moon 
and the time of high water, is nearly constant at the same 
place. It is about fifteen minutes less at the syzigies than 
at the quadratures; but whatever the mean interval is at 
any place, it is called the establishment of the port. 

It is high water at Hudson, on the Hudson river, before The tides 
it is high water at New York, on the same day; but the tide stantly cease 
wave that makes high water one day at Hudson, made high on the remo- 
water at New York the day before ; and the tide waves that va 

•> causes. 

make high water now, were, probably, raised in the ocean 
several days ago ; and the tides would not instantly cease on 
the annihilation of the sun and moon. 

The actual rise of the tide is very different in different Tides ver y 
places, being greatly influenced by local circumstances, such ed ° b ^J 
as the distance and direction to the main ocean, the shape circum- 
of the bay or river, &c, &c. stances - 

In the Bay of Fundy the tide is sometimes fifty and sixty 
feet ; in the Pacific ocean it is about two feet ; and in some 
places in the West Indies, it is scarcely fifteen inches. In 
inland seas and lakes there are no tides, because the moon's 
attraction is equal over their whole extent of surface. 

The following table shows the hight of the tides at the 
most important points along the coast of the United States, 
as ascertained by recent observation. 

Feet. 

Annapolis (Bay of Fundy), 60 

Apple River, 50 

Chicneito Bay (north part of the Bay of Fundy), 60 

Passamaquoddy River, 25 

Penobscot River, 10 

Boston, 11 

Providence, R. I., 5 

New Bedford, 5 

New Haven, 8 

New York, 5 

Cape May, 6 

Cape Henry, 4)^ 

K 



206 ASTRONOMY. 

CHAPTER VI. 

PLANETARY PERTURBATIONS. 

Chap, vi . ^ 192.) The perturbations of a planet, produced by the at- 
pianetary tractions of another planet, are precisely analogous to the per- 

and lunar . 1 r' ' -i'-i l • i 

perturba- turbations oi the moon, produced by the action ot the sun. 

tions anaio- The disturbing forces are of the same kind, and they are 
subject to similar variations from precisely the same causes. 
But the amount of the disturbances is, in most cases, very 
trifling, on account of the small mass of the disturbing planet 
compared with the mass of the sun, or its great distance from 
the body disturbed. 
Action and As action and reaction are everywhere equal, the planets 

on? the plan- mu t ua Uy disturb each other, and if one is accelerated in its 

ets recipro- motion, the other must be retarded ; if the tendency of one to- 
ward the sun is diminished, that of the other must be increased. 
Examine Fig. 23, and conceive V, Venus, to be disturbed 
by the attraction of the earth at E, and if the motion of the 
planets is in the direction of VB, it is perfectly clear that 
Venus will be accelerated by the earth, and the earth will be 
retarded by Venus. 
One planet B u t Venus will be more accelerated in its motion than the 

ed while an- eartQ w iH De retarded, for the disturbance at this point is in 

other is re- a line with the motion of Venus, and not in a line with the 
motion of the earth, 
when the After Venus passes conjunction, that is, passes the varying 
line S E, her motion becomes retarded, and the earth's is ac- 
celerated ; but every motion of the earth we ascribe to the sun ; 
and in all modern solar tables, the corrections of the sun's 
longitude corresponding to the action of Venus, Mars, Ju- 

meant by so- P^ sr > $i? moon, &c, are simply the effect that these bodies 

lar perturba. have on the motion of the earth. 

The direct effect of any of these bodies on the position of 
the sun is absolutely insensible. 

The relative disturbances of two planets are reciprocal to 
their masses ; for if one is double in mass of another, the 



action 
changes. 



PLANETARY PERTURBATIONS. 207 

greater mass will move but half as far as the smaller, under Chap. vi. 
their mutual action. But when the amount of disturbance is 
referred to angular motion for its measure, regard must be regularities 
had to the distances of each planet from the sun ; for the indicate the 
same distance on a larger orbit corresponds to a less angle.* p x anetar 
Also, the whole amount of the disturbing force of a superior disturbance 
planet on an inferior will, at times, be a tangental force a , er cer ai: 

* ' ° reductions. 

( Fig. 23 ) ; but the reaction of the inferior planet on the su- 
perior can never be in a tangent directly with, or opposed to, 
the motion of the superior. 

If observations can give the mutual disturbance of any two 
planets, then these circumstances being taken into considera- 
tion, an easy computation will give the relative masses of the 
planets. 

( 193.) As a general result, the attraction of a superior The gene- 
planet on an inferior, is to increase the time of revolution of ral results in 

respect to the 

the inferior, and to maintain it at a greater distance from the times f rev . 
sun than it would otherwise have. The action of the inferior oiution. 
is to diminish the time of revolution of the superior; and 
the general effect is greater than it would be, if the inferior 
planet were constantly situated at the distance of the sun. 
(Art. 185.) 

As an illustration of this truth, we say, that if Venus were 
annihilated, the length of our year, and the times of revolu- 
tion of all its superior planets, would be a little increased, and 
the revolution of Mercury, its inferior planet, would be a lit- 
tle diminished. If Jupiter were annihilated, the times of re- 
volution of all its inferior planets would be a little diminished ; 
for it acts as a radial force to keep them all a little farther 
from the sun. 

( 194.) If the orbits of all the planets were circular, the inequalities 
acceleration in one part of an orbit would be exactly compen- ™* l,c 

* Geometry demonstrates, that, on the average of each revolution, 
the proportion in which this reaction will affect the longitudes of the 
two planets, is that of their masses multiplied by the square roots of 
the major axes of their orbits, inversely; and this result of a very in- 
tricate and curious calculation is fully confirmed by observation. — 
Herschel. 



208 ASTRONOMY. 

chap. vi. sated by the retardation in another ; and in the course of a 
whole revolution, the mean motions of both planets (the dis- 
turber and the disturbed) would be restored, and the errors 
in longitude would destroy each other. But the orbits are 
not circles, and it is only in certain very rare occurrences 
that symmetry on each side of the line of conjunctions takes 
place ; and hence, in a single revolution the acceleration of 
cds of ine- one P ar ^ cannot be exactly counterbalanced by the retarda- 
^uaiities de- tion of the other; and, therefore, there is commonly left a cer- 
coirunctions * a * n outstanding error, which increases during every synodi- 
in the same cal revolution of the two planets, until the conjunctions take 
parts, o t e pj ace ' m pp 0S it e parts of the orbits, then it attains its maxi- 
mum, which is as gradually frittered away as the line of con- 
junctions works round to the same point as at first. 
Some of Hence, between every two disturbing planets there is a common 
qualities too ^ ne Q ua ^ly depending on their mutual conjunctions, in the same, 
minute to be or nearly in the sa?ne, parts of their orbits. But it would be 
folly to compute the inequalities for every two planets, by rea- 
son of the extreme minuteness of the amounts ; for instance, 
Mercury is not sensibly disturbed by Saturn or Uranus; and 
Mars, and Mercury, and Uranus, practically speaking, do not 
disturb each other; but Jupiter and Saturn have very con- 
siderable mutual perturbations, on account of their orbits be- 
ing near each other, and both bodies far away from the sun. 
The effect (195,) Again, if the revolutions of two planets are ex- 
s te rev0 _ actly commensurate with each other, or, what is the same 
lations of the thing, the mean motion of both exactly commensurate with 
planets. ^ e c i rc i ej ^hen th e conjunctions of those two planets will al- 
ways occur at the same points of the orbits ( just as the con- 
junctions of the two hands of a clock always occur at the 
same points on the dial plate), and, in that case, the conjunc- 
tions will not revolve and distribute themselves around the 
orbits, so that in time, the radial and tangental forces will 
have an opportunity to accelerate on one side of the line of 
conjunctions as much as they retard on the other; and, 
therefore, a permanent derangement would then take place. 
a supposed -p or j ns fc aiice if three times the mean angular motion of 

case for illus- 

tration, one planet were exactly equal to twice the mean angular mo- 



PLANETARY PERTURBATIONS. 209 

tion of another, then three revolutions of the one would ex- Chap. vi. 
actly correspond to two of the other, and every second con- 
junction of the two would take place in the same points of 
the orbits; and the orbits, not being circular, the portions of 
them on each side of the line of conjunctions cannot be sym- 
metrical, unless the longer axes of the two orbits are in the 
same line, and the conjunctions also taking place on that line. 

Here, then, is a case showing that the disturbing force 
may constantly differ in amount on each side of the line of 
conjunctions, and, of course, could never compensate each 
other, and a permanent derangement of these two planets 
would be the result. 

Hence, we perceive, that, to preserve the solar system, it stability of 
is necessary that the orbits should be circles, or their times thesolars y s ' 

. tem. 

of revolution incommensurable ; but we do not pretend to say 
that the converse of this is true ; we do say, however, that no 
natural cause of destruction has thus far been found. 

( 196.) The times of the planetary revolutions are incom- 
mensurable ; but, nevertheless, there are instances that ap- 
proach commensurability, and, in consequence, approach a 
derangement in motion, which, when followed out, produce 
very long periods of inequality, called secular variation. The 
most remarkable of these, and one which very much perplexed 
the astronomers of the last century, is known by the term of 
" the great inequality " of Jupiter and Saturn. 

"It had long been remarked by astronomers that, on com- The great 
paring together ancient with modern observations of Jupiter 1 » e( i 1iallties 
and Saturn, their mean motions could not be uniform." The and Saturn. 
period of Saturn appeared to have been increased throughout 
the whole of the seventeenth century, and that of Jupiter 
shortened. Saturn was constantly lagging behind its calcu- 
lated place, and Jupiter was as constantly in advance of his. 
On the other hand, in the eighteenth century, a process pre- 
cisely the reverse was going on. 

The amount of retardations and accelerations, corresponding The per- 
to one, two, or three revolutions were not very great ; but, as f**** g ™ 
they went on accumulating, material differences, at length, sopherg. 
existed between the observed and calculated places of both 
14 R* 



210 ASTRONOMY. 

Chap. vi. these planets, and, as such differences could not then be ac- 
counted for, they excited a high degree of attention, and 
formed the subject of prize problems of several philosophical 
societies. 
Laplace For a long time these astonishing facts baffled every en- 
1 e deavor to account for them, and some were on the point of 

mystery. * 

declaring the doctrine of universal gravity overthrown ; but, 
at length, the immortal Laplace came forward, and showed 
the cause of these discrepancies to be in the near commensu- 
rability of the mean motions of Jupiter and Saturn; which 
cause we now endeavor to bring to the mind of the reader in 
a clear and emphatic manner. 

( 197.) The orbits of both Jupiter and Saturn are ellipti- 
cal, and their perihelion points have different longitudes, and, 
therefore, their different points of conjunction are at different 
distances from each other, and no line* of conjunction cuts the 
two orbits into two equal or symmetrical parts ; hence, the 
inequalities of a single synodical revolution will not destroy 
each other ; and, to bring about an equality of perturbations, 
requires a certain period or succession of conjunctions, as we 
are about to explain. 
The revo- Five revolutions of Jupiter require 21663 days, and two 
utionsofju- of gat 21518 days. So that, in a period of two revolu- 

,>iter and Sa- J L 

aim compar- tions of Saturn (about sixty of our years), after any conjunc- 
tion of these two planets, they will be in conjunction again not 
many degrees from where the former took place. 
Their syno- To determine definitely where the third mean conjunction 

dical revolu- w -^ £ a k e pi aC6) we compute the synodical revolution of these 

tion deter- ■ . . .,. 

mined. two planets by dividing the circumference of the circle in sec- 
onds (1296000) by the difference of the mean daily motion 
of the planets in seconds (178". 6), f and the quotient is 7253.4 
days ; three times this period is 21760 days. In this period 
Jupiter performs five revolutions and 8° 6' over; Saturn 
makes two revolutions and 8° 6' over ; showing that the line 

* Line of conjunction, an imaginary line drawn from the sun 
through the two planets when in conjunction. 

f See problem of the two couriers, Robinson's Algebra. 



PLANETARY PERTURBATIONS 



211 



of conjunction advances 8° 6' in longitude during the period Chap. vi. 
of 21760 days. 

In the year 1800, the longitude of Jupiter's perihelion point 
was 11° 8', and that of Saturn 89° 9'; the inclination of the 
greater axis of the orbits, therefore, was 78° 1'. 

Fig. 42. 




plained. 



Let AB ( Fig. 42 ) represent the major axis of Saturn's The series 
orbit, and a b that of Jupiter; the two are placed at an angle tions 00 ^ 

Of 78°.* plained. 

Suppose any conjunction to take place in any part of the 
orbits, as at JS (the line JS we call the line of conjunc- Lineofcon- 
tion) ; in 7253.4 days afterward another conjunction will take J™ ctl ° n 
place. In this interval, however, Saturn will describe about 
243° in its orbit^ at a mean rate, and Jupiter will describe one 
revolution and about 243° over, and it will take place as re- 
presented in the figure, at P Q ( S TB being the direction of 
the motion). The next conjunction will be 243° from PQ, or 
at R T. From RT the next conjunction will be at si, 8° 6' 
in advance of JS, and thus the conjunction JS ( so to speak) 
will gradually advance along on the orbit from S to T. 

But, as we perceive, by inspecting the figure, there is a 

* We have very much exaggerated the eccentricities of these ellip- 
ses, for the purpose of magnifying the principle under consideration. 



212 



ASTRONOMY. 



Chap. VI. 

Certain 
conjunctions 
■ bring the pla- 
nets nearer 
together than 
most others. 



The period of 
this remark- 
able ine- 
quality com- 
puted, and 
the computa- 
tion confirm- 
ed by obser- 
vation. 



An expla- 
nation of the 
principle that 
led to the 
discovery of 
Neptune. 



certain portion of the orbits, between S and T, where the two 
planets would come nearer together in their conjunction, than 
they do at conjunctions generally, and, of course, while any 
one of the three conjunctions is passing through that portion 
of the orbits — Jupiter disturbs Saturn, and Saturn reacts on 
Jupiter more powerfully than at other conjunctions ; and this 
is the cause of "the great inequality of Jupiter and Saturn" 

( 198. ) To obtain the period of this inequality, we com- 
pute the time requisite for one of these lines of conjunction 
to make a third of a revolution, that is, divide 120° by 8° 6', 
and we shall find a quotient of 14f f , showing the period to be 
14|^ times 21760 days, or nearly 883 years; which would be 
the actual period, provided the elements of the orbits re- 
mained unchanged during that time. But in so long a period 
the relative position of the perigee points will undergo con- 
siderable variation ; which causes the period to lengthen to 
about 918 years. 

The maximum amount 
of this inequality, for 
the longitude of Saturn, 
is 49', and for Jupiter 
21', always opposite in 
effect, on the principle of 
action and reaction. 

(199.) The last great 
achievement of the pow- 
ers of mind in the solar 
system, was the discovery 
of the new planet Nep- 
tune, by Leverrier and 
Adams analyzing the in- 
equalities of the motion 
of Uranus. To give a rude explanation of the possibility of 
this problem we present Figure 43. Let S be the sun, and 
the regular curve the orbit of Uranus, as corresponding to all 
known perturbations; but at a it departs from its computed 
track and runs out in the protuberance acb. This indicated 
that some attracting body must be somewhere in the direction 




ABERRATION. 21o 

S P, although no such body was eyer seen or known to exist. Cha? - y l 
The next time the planet comes round into the same portions 
of its orbit,* suppose the center of the protuberance to have 
changed to the line S Q. This would indicate that the un- How com ' 

• >"<, putations 

known and unseen body was now in the line S Q, and that CO uid be 
since the former observations it had changed positions by the made for ths 
angle P S Q; and, by this angle, and the time of its descrip- an unseeil 
tion, something like a guess could be made of the time of its planet, 
revolution. 

With the approximate time of revolution, and the help of 
Kepler's third law, its corresponding distance from the sun 
can be known. With the distance of the unseen body, and 
the amount that Uranus is drawn from its orbit by it, we can 
approximate to its mass. 

Thus, we perceive, that it is possible to know much about 
an existing planet, although so distant as never to be seen. 
But the body that disturbed the motion of Uranus has been 
seen, and is called Neptune. 



Chap. VII, 



CHAPTER VII. 

ABERRATION, NUTATION, AND PRECESSION OE THE EQUINOXES. 

(200.) About the year 1725 Dr. Bradley, of the Green- 
wich observatory, commenced a very rigid course of observa- l , s bs^r- 
tions on the fixed stars, with the hope of detecting their rations on 
parallax. These observations disclosed the fact, that all the lhe fixed , 

1 .... stars ior tua 

stars which come to the upper meridian near midnight, have purpose of 
an inverse of longitude of about 20", while those opposite, findin s their 
near the meridian of the sun, have a decrease of longitude of unexpected 
20" ; thus making an annual displacement of 40". These resnits. 
observations were continued for several years, and found to 
be the same at the same time each year ; and, what was most 

♦Leverrier and Adams had not the advantage of a complete revolu- 
tion of Uranus. 



214 



ASTRONOMY, 



chap. vii. perplexing, the results were directly opposite from such as 
would arise from parallax. 

These facts were thrown to the world as a problem demand- 
ing solution, and, for some time, it baffled all attempts at ex- 
planation, but it finally occurred to the mind of the Doctor, 
that it might be an effect produced by the progressive motion 
of light combined with the motion of the earth ; and, on strict 
examination, this was found to be a satisfactory solution. 



Fig. 

Aberration 


44. 


( 201.) A person stand- 


illustrated. S 

* * 


ing still in a rain shower, 


when the rain falls perpen- 






dicularly, the drops will 


\ 

\ 




strike directly on the top 




of his head; but if he 


\ 




starts and runs in any di- 


\ 
\ 




rection, the drops will strike 


\ 
\ 




him in the face ; and the 


\ 
\ 




effect would be the same, 


\ 
\ 




in relation to the direction 


\ 




of the drops, as if the per- 


\ 




son stood still and the rain 


\ 

\ 




came inclined from the di- 




rection he ran. 




D 


This is a full illustration 


\\ \ 




of the principle of these 
changes in the positions 
of the stars, which is called 
Aberration; but the follow- 
ing explanation is more 
appropriate. 


Another and \ \ 




Conceive the rays of 


more appro* \ \ 
priate illus- \ > 


L \ 


light to be of a material 


tTS.t.inn \ 


\ \ 


\ substance, and its nartinles 


1 


3 A progressive, passing from 



the star S (Fig. 44) to the earth at B; passing directly 
through the telescope, while the telescope itself moves from 
A to B by the motion of the earth. And if DB is the mo- 
tion of light, and A B the motion of the earth, then the tele- 



ABERRATION 



215 



scope must be inclined in the direction of A D, to receive the Chaf - vn 
light of the star, and the apparent place of the star would be 
at S', and its true place at S, and the angle ADB is 20 ".36, at 
its maximum, called the angle of aberration. 

By the known motion of the earth in its orbit, we have the 
value of A B corresponding to one second of time : we have 
the angle A D B by observation : the angle at B, is a right 
angle, and (from these data) computing the side BD we 
have the velocity of light, corresponding to one second of 
time. To make the computation, we have 

DB:BA:: Rod. : tan. 20".36.* 

But B A, the distance which the earth moves in its orbit The veio- 

Fig. 45. cit y of H s ht 

q^ computed by 

means of ab- 
*% 7v erration. 



180 * 




* 



270 



*To obtain the logarithmetic tangent of 20'\36 see note on page 128. 

15 



216 ASTRONOMY. 

Chap. vii. in one second of time, is within a very small fraction of 19 
miles; the logarithm of the distance is 1.278802, and, from 
this, we find that BD must be 192600 miles, the velocity of 
light in a second ; a result very nearly the same as before 
deduced from observations on the eclipses of Jupiter's moons. 
(Art. 143.) 

The agreement of these two methods, so disconnected and 
so widely different, in disclosing such a far-hidden and re- 
markable truth, is a striking illustration of the power of 
science, and the order, harmony, and sublimity that pervades 
the universe. 
a compre. To show the effects of aberration on the whole starry 
ofthHffe e tI neavens > we g* ve figure 45. Conceive the earth to be 
of aberra- moving in its orbit from A to B. The stars in the line AB, 
tion. whether at or 180, are not affected by aberration. The 

stars, at right angles to the line A B, are most affected by 
aberration, and it is obvious that the general effect of aberra- 
tion is to give the stars an apparent inclination to that part 
of the heavens, toward which the earth is moving. Thus 
the star at 90 has its longitude increased, and the star op- 
posite to it, at 270, has its longitude decreased, by the effect 
of aberration; both being thrown more toward 180. The ef- 
fect on each star is 20". 36. But when the earth is in the 
opposite part of its orbit, and moving the other way, from C 
to D, then the star at 90 is apparently thrown nearer to ; 
so also is the star at 270, and the whole annual variation 
of each star, in respect to longitude, is 40".72. 
Proof ofthe / 202. ) The supposition of the earth's annual motion fully 

annual mo- _ . 

tion of the explains aberration; conversely, then, the observed variations 
earth. f the stars, called aberration, are decided proof s ofthe earth's 

annual motion. 

In consequence of aberration, each star appears to describe 
a small ellipse in the heavens, whose semi-major axis is 20".36, 
and semi-minor axis is 20". 36 multiplied by the sine of the 
latitude of the star. The true place of the star is the center 
of the ellipse. If the star is on the ecliptic, the ellipse, just 
mentioned, becomes a straight line of 40". 72 in length 
If the star is at either pole of the ecliptic, the ellipse be- 



ABERRATION. 217 

comes a circle of 40". 72 in diameter, in respect to a great Chap, vi i 
circle ; but a circle, however small, around the pole, will in- 
clude all degrees of longitude ; hence it is possible for stars 
very near either pole of the ecliptic, to change longitude 
very considerably, each year, by the effect of aberration ; but 
no star is sufficiently near the pole to cause an apparent revo- 
lution round the pole by aberration ; and the same is true in 
relation to the pole of the celestial equator. 

All these ellipses have their longer axis parallel to the ecliptic, 
and for this reason it is easy to compute the aberration of a 
star in latitude and longitude,* but it is a far more complex 
problem to compute the effects in respect to right ascension 
and declination. 

( 203. ) The aberration of the sun varies but a very little, Aberration 
because the distance to the sun varies but little, and without 
material error, it may be always taken at 20". 2, subtractive. 
The apparent place of the sun is always behind its true place 
by the whole amount of aberration ; but the solar tables give 
its apparent place, which is the position generally wanted. 

In computing the effect of aberration on a planet, regard 
must be had to the apparent motion of the planet while light 
is passing from it to the earth. 

The effects of aberration on the moon are too small to be The moon 
noticed, as light passes that distance in about one second of mot affected 

by aberra- 
time - tion. 

( 204. ) While Dr. Bradley was continuing his observa- other ine- 
tions to verify his theory of aberration, he observed other qua ^^ ° b " 

J r 7 served by Dr. 

small variations, in the latitudes and declinations of the stars, Bradley, 
that could not be accounted for on the principle of ab- 
erration. 

The period of these variations was observed to be about 

_20".36cos.(#— s) 

*Aber. m Lon. = ^ ; 

cos. I 

Aber. in Lat. = 20". 36 sin. (S—s) sin. I. 

In these expressions S represents the longitude of the sun, 
s the longitude of the star, and I its latitude. . 

s 



218 



ASTRONOMY. 



Chap, vi i. the same as the revolution of the moon's node, and the 
amount of the variation corresponded with particular situa- 
tions of the node ; and, in short, it was soon discovered that 
the cause of these variations was a slight vibration in the 
earth's axis, caused by the action and reaction of the sun and 
moon on the protuberant mass of matter about the equa- 
tor, which gives the earth its spheroidal form, and the effect 
itself is called Nutation. 

Fig. 46. 



m 




* 
# * * 
* * * 
* 



Nutation / 205. ) We have shown, in Art. 176, that the attraction 

fully explain- (,-,-, . 

edbythethe- °* a body, m, on a ring of matter around a sphere, has the 
ory of gravi- effect of making the plane of the ring incline toward the at- 
y ' tracting body. 

Let B O, Fig. 46, represent the plane of the equator ; and 
conceive the protuberant mass of matter, around the equator, 
to be represented by a ring, as in the figure. Let m be the 



NUTATION. 219 

moon at its greatest declination, and, of course, without the Chap, vn. 
plane of the ring. 

Let P be the polar star. The attraction of m on the ring 
inclines it to the moon, and causes it to have a slight motion 
on its center ; hut the motion of this ring is the motion of the 
whole earth, which must cause the earth's axis to change its 
position in relation to the star P, and in relation to all the 
stars. 

When the moon is on the other side of the ring, that is, 
opposite in declination, the effect is to incline the equator to 
the opposite direction, which must be, and is, indicated by an 
apparent motion of all the stars. 

A slight alternate motion of all the stars in declination, cor- 
responding to the declinations of the sun and moon, was care- 
fully noted by Dr. Bradley, and since his time has been fully 
verified and definitely settled ; this vibratory motion is 
known by the name of nutation, and it is fully and satisfac- 
torily explained on the principles of universal gravity ; and 
conversely, these minute and delicate facts, so accurately and 
completely conforming to the theory of gravity, served as one 
of the many strong points of evidence to establish the truth 
of that theory. 

( 206.) By inspecting Fig. 46, it will be perceived that The e ene - 
when the sun and moon have their greatest northern declina- nTltati(m a. 
tions, all the stars north of the equator and in the same hemi- lustrated by 
sphere as these bodies, will incline toward the equator; or all Fl s- 46 - 
the stars in that hemisphere will incline southward, and those 
in the opposite hemisphere will incline northward ; the amount 
of vibration of the axis of the earth is only 9".6 (as is shown 
by the motion of the stars), and its period is" 18.6, or about 
nineteen years ; the time corresponding to the revolution of 
the moon's node. When the moon is in the plane of the 
equator, its attraction can have no influence in changing the 
position of that plane ; and it is evident that the greatest ef- 
feet must be when the declination is greatest. node must be 

The moon's declination is greatest when the longitude of tocorrespond 

i , ,. t rT n i> a • tothemoon's 

the moon s ascending node is 0, or at the first point ot Aries. grea test de- 
The greatest declination is then 28° on each side of the ciinatum. 



220 ASTRONOMY. 

chap. vii. equator ; but when the descending node is in the same point, 
the moon's greatest declination is only 18°. Hence there will 
be times, a succession of years, when the moon's action on the 
protuberant matter about the equator must be greater than in 
an opposite succession of years, when the node is in an oppo- 
site position. Hence, the amount of lunar nutation depends 
on the position of the moon's nodes. 

Monthly mi- it j s ver y natural to suppose that the period of lunar nuta- 

smaii.' ti° n wou ld De simply the time of the revolution of the moon ; 
and so, in fact, it is ; but the corresponding amount is very 
small, only about one-tenth of a second. This is because half 
a lunar revolution, about 13i days, while the moon is on one 
side of the equator, is not a sufficient length of time for the 
moon to effect much more than to overcome the inertia of the 
earth ; but, in the space of nine years, effecting a little more 
than a mean result at every revolution, the amount can rise to 
9". 6, a perceptible and measurable quantity. 
The mean (207.) The mean course of the moon is along the ecliptic; 

effect of the jj. g var i a ti on from that line is only about five degrees on each 

moon on the # . 

of mat- side; hence, the medium effect of the moon on the protuberant 
around niass of matter at the equator is the same as though the 
moon was all the while in the ecliptic. But, in that case, its 
effect would be the same at every revolution of the moon ; 
and the earth's equator and axis would then have an equili- 
brium opposition, and there would be no nutation, save the 
slight monthly nutation just mentioned, which is too small to 
be sensible to observation ; and the nutation that we observe, 
is only an inequality of the moon's attraction on the protube- 
rant equatorial ring ; and, however great that attraction might 
be, it would cause no vibration in the position of the earth, 
if it were constantly the same. 
Solar nu- We have, thus far, made particular mention of the moon, 
tauon. i^ there is also a solar nutation; its period is, of course, a 
year ; and it is very trifling in amount, because the sun at- 
tracts all parts of the earth nearly alike; and the short 
period of one year, or half a year (which is the time that the 
unequal attraction tends to change the plane of the ring in 



mass 

ter 

the equator 



THE EQUINOXES. 221 

one direction^, is too short a time to have any great effect on Chap. vn. 
the inertia of the earth. 

The solar nutation, in respect to declination, is only one 
second. 

( 208.) Hitherto we have considered only one effect of nu- 
tation — that which changes the position of the plane of the 
equator — or, what is the same thing, that which changes the 
position of the earth's axis ; but there is another effect, of 
greater magnitude, earlier discovered, and better known, re- 
sulting from the same physical cause, we mean the 

PRECESSION OF THE EQUINOXES. 

We again return to first principles, and consider the mu- First prin- 
tual attraction between a ring of matter and a body situated ciples agam 

° ^ examined. 

out of the plane of the ring ; the effect, as we have several 
times shown, is to incline the ring to the body, or (which is 
the same in respect to relative positions), the body inclines 
to run to the plane of the ring. 

The mean attraction of the moon is in the plane of the The mean 
ecliptic. The sun is all the while in the ecliptic. Hence, the the sun Md 
mean attraction of both sun and moon is in one plane, the moon are in 
ecliptic ; but the equator, considered as a ring of matter sur- °" e p ane ' 
rounding a sphere, is inclined to the plane of the ecliptic by 
an angle of 23i degrees, and hence, the sun and moon have a 
constant tendency to draw the equator to the ecliptic, and 
actually do draw it to that plane ; and the visible effect is, 
to make both sun and moon, in revolutions, cross the equator 
sooner than they otherwise would, and thus the equinox falls 
back on the ecliptic, called the precession of the equinoxes. 

The annual mean precession of the equinoxes is 50". 1 of Thepreces- 

1 , , , . sion of the 

arc, as is shown by the sun coming into the equinox, or equinoxes, 
crossing the equator at a point 50".l, before it makes a revo- 
lution in respect to the stars. 

Perhaps it is clearer to the mind to say, that the sun is Natural 
drawn to the equator by the protuberant mass of matter sion# 
around the earth, and, in consequence, arrives at the equator, 
in its apparent revolutions, sooner than it otherwise would. 
But the truth is, that the ecliptic is stationary in position, 

s* 



222 ASTRONOMY. 

chap. vii. and the equator, by a slight motion, meets the ecliptic ; which 
motion is caused by the attractions of the sun and moon, as 
has been several times explained. 
The tme -j-£ ^ e moon were a pj fae while in the ecliptic, the preces- 

pnysical *■ . - *■ 

cause of the sion of the equinoxes would then be a constantly flowing quan- 
precession of ^ e q Ua i to 50". 1, for each year; but, for a succession of 

the equinox- . 

es . about nine years, the moon runs out to a greater declination 

than the ecliptic, and, during that time, its action on the 
equatorial matter is greater than the mean action, and then 
comes a succession of about nine years, when its action is 
less than its mean ; hence, for nine years, the precession of 
the equinoxes will be more than 50". 1, per year, and, for the 
nine years following, the precession will be less than 50". 1, 
for each year ; and the whole amount of variation, for this in- 
equality, in respect to longitude, is 17". 3, and its period is half 
a revolution of the moon's nodes. This inequality is called 
the equation of the equinoxes, and varies as the sine of the 
longitude of the moon's nodes. 
Equation The equation of the equinoxes, of course, affects the length 
equl " of the tropical year, and slightly, very slightly, affects side- 
real time. 

Mean and There is a true equinox and a mean equinox; and, as side- 
true sidereal i ■• • i <? .i «v ± «i o ,1 

real time is measured trom the meridian transit of the equi- 

time. J- 

nox, there must be a true sidereal and a mean sidereal time ; 
but the difference is never more than 1.1 s. in time, and, gene- 
rally, it is much less. 
Explanation ( 209.) In the hope of being more clear than some authors 
lg ' have been, in explaining the results of precession, we present 
Fig. 47. i? represents the pole of the ecliptic, and the great 
circle around it is the ecliptic itself. P is the pole of the 
earth, 23° 27' from the pole E, and around P, as a center, we 
have attempted to represent the equator, but this, of course, 
is a little distorted; qp and &= are the two opposite points 
where the ecliptic and equator intersect; °pE is the first me- 
ridian of longitude; cpP is the first meridian of right ascen- 
sion. The angle E^pP is 23° 27', and E P, produced, is the 
meridian passing through the solstitial points. To obtain a 
clear conception of the precession of the equinoxes, the stars, 



THE EQUINOXES. 



223 



the ecliptic, and its pole E, must be considered as fixed, chap, vii 
and the line °p ^= as having a slow motion of 50". 1, per an- 

Fig. 47. 




. i i • From the 

nam, on the ecliptic, in a retrograde direction; and this must fixed pogi . 
carry the pole P, around the point E, as a center, carrying tion of the 
also the solstitial points backward on the ecliptic. Some also ' f the 
of the stars have proper motions ; but, putting that circum- stars, the 
stance out of the question, the stars are fixed, and the eclip- s * ars never 

, change lati- 

tic is fixed ; therefore, the stars never change latitude, but tude. 



224 ASTRONOMY. 

chap, vii. the whole frame-work of meridians from the pole P, the pole 
itself, and the equator, revolve over the stars ; and, in respect 
to that motion of the meridian and the equator, the stars 
change rigid ascension, declination, and longitude, hut do not 
change latitude. The stars change longitude, simply because 
the first meridian of longitude, T E, moves backward ; they 
change right ascension, because the meridian, qp P, and all 
the meridians of right ascension, revolve backward. 
One hemi- By inspecting the figure, we readily perceive that all the 
stars ap- s ^ ars near HP must, apparently, approach the north pole, be- 
pro aches the cause the pole, in its revolution round E, is approaching to- 
the other V' war ^- ^ Da ^ P ar ^ °f * ne ecliptic ; for the same reason, all the 
cedes from stars near =£= are, apparently, moving southward, because the 
equator is being drawn over them. In short, all the stars, 
from the eighteenth hour of right ascension through qp, to 
the sixth hour of right ascension, must diminish in north po- 
lar distance, and all the stars, from the six hours through =£=, 
to the eighteenth hour of right ascension, must increase in 
north polar distance, in consequence of the precession of the 
equinoxes, 
inspection These observations may be confirmed by inspecting Table 
* II, in which is registered the positions of the principal fixed 
stars, with their annual variations. The column of annual 
variation of declination changes sign at the point correspond- 
ing to six hours, and eighteen hours of right ascension ; and 
the rapidity of this variation is greater as the star is nearer 
to hours, or twelve hours of right ascension. 
Annual va- When the right ascension of a star is hours, or twelve 
sanation hours, it is easy to compute its annual variation in declina- 
how comput- tion, corresponding to its precession along the ecliptic of 
50". 1. Conceive a small plane triangle whose hypothenuse is 
50".l, the angle at the base 23° 27' 40" (i. e. the obliquity 
of the ecliptic ), the side opposite to this angle will be found 
to be a little over 20", corresponding to the figures in the 
table. 
Proper mo- j^ j s thus, by the motion of these imaginary lines over the 
discovered whole concave of the heavens, that the annual variation of 
both right ascension and declination of each individual star 



THE EQUINOXES. 225 

in the catalogue is computed and put down ; and if any par- chap. vn. 
ticular star does not correspond with this, it is said to have 
proper motion ; and it is thus that proper motions are detected. 

As P must circulate round E by the slow motion of 50". 1 Final effect 
in a year, it will require 25868 years to perform a revolution; 



sion. 



and the reader can perceive, by inspecting the figure, why 
the pole star is in apparent motion in respect to the pole, and 
why that star will cease to be the polar star, and why, at the 
expiration of about 12000 years, the bright star, Lyra, will 
be the polar star. 

(210.) The mean effect of the moon in producing the pre- Compara- 
cession of the equinoxes is, to the mean effect of the sun, as tive effect of 

. , . sun and 

five to two. The sun s action is nearly constant, because moon- 
the sun is always in the ecliptic ; a small annual variation, 
however, is observed. The great inequality of 17". 3, corre- 
sponding to about nineteen years, is caused entirely by the 
unequal action of the moon, depending on the longitude of 
the moon's ascending node. 

In consequence of this inequality, the pole, P, does not unduiatory 
move round the pole of the ecliptic, E, in an even circumfe- motion of the 
rence of a circle, but it has a waving or undulating motion, as a"und "e 
represented in this figure ; each wave pole of the 

corresponding to nineteen years ; and, _r-~'~ s ->-w ecliptic, 

therefore, there must be as many of 
them in the whole circle as 19 is con- 
tained in 25868. From this, we per- 
ceive, that the undulations in the fig- 
ure are much exaggerated, and vastly 
too few in number; an exact linear 
representation of them would be im- 
possible. 

(211.) From the foregoing, we learn that the positions of M ean and 
all the stars are affected by aberration, precession, and nuta- apparent 
tion ; the amount for each cause is very trifling in itself, yet, star 
in most cases, too great to be neglected, when accuracy is 
required ; and it is as difficult to make computations for a 
small quantity as for a large one, and often greater ; and to 
reduce the apparent place of a fixed star from its mean place, 
15 




226 ASTRONOMY. 

Chap. vii. and its mean place from its apparent place, is one of the most 

troublesome problems in practical astronomy. 
General for- The mean place of a fixed star, reduced to the time of ob- 

mulee, where . . ~, . , . , 

found. servation, is sumciently near its apparent place to be con- 

sidered the same. The practical astronomer, however, who 
requires the star as a point of reference, or uses it for the 
adjustment of his instruments, must not omit any cause of 
variation; but such persons will always have the aid of a 
Nautical Almanac, where general formulae and tables will be 
found, to direct and facilitate all the requisite reductions, 
importance ^ 212.) Physical astronomy brings many things to light 

astronomy. ^at would- otherwise escape observation, and some of these 
developments, at first, strike the learner with surprise, and he 
is not always ready to yield his assent. For instance, as a 
general student, he learns that the anomalistic year, the time 
that the earth moves from its perigee to its perigee again, is 
365 d. 6h. 14 m. ; that the perigee is very slow in its motion, 
moves only about 12" in a year, and is subject to but few 
fluctuations. He has also learned that the earth, in its orbit, 
describes equal areas in equal times ; hence, he concludes, 
that the time from perigee to perigee, or from apogee to apo- 
gee, must be very nearly a constant quantity; but, on con- 
sulting and comparing the predictions to be found in the En- 
glish nautical almanacs, he will find these periods to be (in 
comparison to his anticipations) very fluctuatiDg. They 
differ from the stated mean times, not only by minutes and 
seconds, but by hours, and even days. The investigator is, 
at first, surprised, and fancies a mistake ; at least, a mis- 
print ; but, on examining concurrent facts, such as the lo- 
garithms of the distance from the sun, and the sun's true 
motion at the time, he finds that, if a mistake has been made, 
it is a very harmonious one, and every other circumstance has 
been adapted to it. 
The lati- But let U3 turn a moment from these facts, and examine 
, . e the first page of our Tables. There it will be found, that the 

sim explain- i. » » 

ed. sun has latitude ; that it deviates to the north and south of 

the ecliptic, by a quantity too small ever to be observed ; it is, 
therefore, a quantity wholly determined by theory, and, as 



THE EQUINOXES. 



227 



the sun's latitude changes with the latitude of the moon, we Chap, vil 



Fig. 48. 
S 




must seek for its cause in the lunar motions. 

To understand the fact of the sun having 
latitude, we must admit that it is the center 
of gravity "between the earth and moon, that 
moves in an elliptical orbit round the sun; 
and that center is always in the ecliptic ; and 
the sun, viewed from that point, would have 
no latitude. But when the moon, m, (Fig. 
48 ), is on one side of the plane of the eclip- 
tic, Sc, the earth, E would be on the other m 
side, and the sun, seen from the center of the 
earth, would appear to lie on the same side 
of the ecliptic as the moon. Hence, the sun 
will change his latitude, when the moon changes 
her latitude. 

If the moon were all the while in the plane of the ecliptic, 
the sun would have no latitude ( save some extremely minute 
quantities, from the action of the planets, when not in the 
plane of the ecliptic ) ; but the moon does not deviate more 
than 5° 20 from the ecliptic, and, of course, the earth makes 
but a proportional deviation on the other side ; but, in longi- 
tude, the moon deviates to a right angle on both sides, in re- 
spect to the sun, and when the moon is in advance in respect 
to longitude, the sun appears to be in advance also; and 
when the moon is at her third quarter, the longitude of the 
sun is apparently thrown back by her influence : — the great- 
est variation in the sun's longitude, arising from the motion 
of the earth and moon about their center of gravity, is about 
6" each side of the mean. Now it is this motion of the 
earth around the common center of gravity of the earth and 
moon, that chiefly affects the time when the earth comes to 
its apogee and perigee. When the moon is in conjunction 
with the sun, the center of the earth is farther from the sun 
than it otherwise would be ; and when the moon is in oppo- 
sition to the sun, the earth is about 3200 miles nearer the 
sun than it would be in its mean orbit ; and thus, we per- 
ceive, that the longitude of the moon has a great influence in 



Longitude 
of the sun af- 
fected by the 
position of 
the moon. 



Longitude 
of the moon 
affects the 
time that the 
earth comes 
to its apogee 
and perigee. 



228 ASTRONOMY. 

m 

chap. vu. bringing the earth into, or preventing it from coming into, its 

perigee or apogee; but the perigee and apogee points, for the 
center of gravity, are quite uniform, agreeably to the views ex- 
pressed in the first part of this article. These explanations 
will give a general insight into some of the apparent intrica- 
cies of physical astronomy. 
Small equa- The small equations of the sun's center are computed on 
nons of the ^ e principle explained by Fig. 48, the sun having a mo- 
expiained. tion round the center of gravity between itself and each of 
the planets. For example, the perturbation produced by Ju- 
piter is greatest when Jupiter is in longitude 90° from the 
sun, as seen from the earth ; the greatest effect is then about 
8", and varies very nearly as the sine of Jupiter's elongation 
from the sun. 

When Jupiter is in conjunction with the sun, the sun is 
nearer the earth than it otherwise would be, and, on this ac- 
count, we have a small table to correct the sun's distance 
from the earth, called the perturbations of the sun's distance. 
The same remarks apply to other planets, but, to avoid 
confusion, the effects of each one must be computed sepa- 
rately. 



PRACTICAL ASTRONOMY. 229 

SECTION IV. 
PEACTICAL ASTRONOMY. 



PREPARATORY REMARKS. 

We have now done with general demonstrations, and with 
minute and consecutive explanations; but we shall give all 
necessary elucidation in relation to the particular problems 
under consideration. To go through this part of astronomy 
with success and satisfaction, the reader must have a passa- 
ble understanding of plane and spherical trigonometry ; and 
if to these he adds a general knowledge of the solar system, 
as taught in the foregoing pages, he will have a full compre- 
hension of all we design to embrace in this section. 

To prompt the student in his knowledge of trigonometry 
we give the following formulae : 

I. Relative to a single arc or angle. 

1. - - - sin. a = tan. a cos. a.* 

tan. a 

2. - - - sin. a 



3. - - - cos. a = 



v l-(-tan. 2 a. * 
1 



Vl-(-tan. 2 a. 

4. - - - cos. a = 2 cos. 2 \ a — 1. 

- sin. a 

o. - - - tan. \a=- 



1-f-cos. a 

a o i 1 — cos - a 

6. - - - tan. 2 ia=— . 

l-}-cos. a 

7. - - - sin. 2a=2 sin. a cos. a. 



Trig. 



* Radius is unity in all these equations. 



230 ASTRONOMY. 

Two. 8. - cos. 2a=2 cos. 2 a — 1=1 — 2 sin. 2 a. 

II. Relative to two arcs, a and b, of which a is supposed 
to be the greater. 

9. - sin. (a-)-5)=sin. a cos. 5-)-sin. b cos. a. 

10. - cos. ('«-f-5)=cos. a cos. b — sin. a sin. b. 

11. - sin. (a — 5)=sin.a cos. b — sin. b cos. a. 

12. - cos. (a — 5)=cos. a cos. 5-j-sin. a sin. 5. 
Sum of (9 ) and ( 11 ) gives 13, diff. gives 14. 

13. - sin. («— |— 5)— [— sin. (a — 5) =2 sin. a cos. b. 

14. - sin. (a-\-b) — sin. (a — b) =2 cos. a sin. 5. 

^ e ,',•■;"» tan. am- tan. 5 

15. - tan. («-H) - == 7-. 

1 — tan.a tan. b. 

Hr , . _ x tan.a — tan. b 

lo. - tan. (a — o) - =t~, r- 

1-4-tan. a tan. o 



17. - 

18. - 

t 

19. \ 



sin. a_f_sin. 5 __tan. ± (a-\-b) 

sin. a — sin. b tan.i(a — b)' 

tan. a-4-tan. 5 _sin. (a-\-b) 

tan.a — tan. 5 sin. (a — &)* 

1+tan. & ..ro, A 

- =tan. (45°-f-6), 

1— tan. b v y 



1 — tan. b 

{ i+sju " =tan - ( 45a -^ 

We shall, probably, make an application of the following 
theorem ; it applies to finding the unknown angles of a tri- 
angle, when the logarithms of two sides (not the sides them- 
selves) and the angle included between the sides are given. 

The greater of two sides of a plane triangle is, to the less, 
as radius to the tangent of a certain angle. Take this angle 
from 45°, and call the difference a. Lastly, radius is to the 
tangent, a, as the tangent of the half sum of the angles at the 
base is to the tangent of half their difference. 

III. Resolution of right-angled spherical triangles. 

In the following equations, h is the hypothenuse, s a given 



EQUATIONS 



231 



side, a a given angle, and x the quantity sought. (The right trig. 
angle is unity, and always given.) 




Required, 
side op. a 
side adj. a 
the other angle 



Solution. 

20. sin. #=sin. k sin. a. 

21. tan. £=tan. h cos. a. 

22. cot. #=cos. h tan. a. 



h 



the other side 23. cos. x= 



cos. h 



and «{ ang. adj. s 

s 

ang. op. s 



s 

and 
a 
opposite, 



cos. s 

24. cos. #=tan. s cot. k 

. sin. s 

25. sin. x=- — ; 
sin. k 

. sin. s 

h 2o. sin. #==-: 

sin. a 

27. sin. #=tan. s cot. a 

cos. a 



the other side 

the other ang. 28. sin. x 



cos. s 



s 
and 



I h . 

1 the other side, 
& I 

- the other ang. 

adjacent, I 



29. cot. #=cos. a cot. s 

30. tan. #=tan. a sin. 5 

31. cos. #=sin. a cos. 5. 



The ( h 

two sides. ( the angles, 



32. cos. #=cos. s cos. other side 

33. cot. #=sin. adj. side X cot. 

[opp. side. 

IY. Resolution of oblique angled spherical triangles. 
Let A B and C be the three angles of any spherical triangle, 
and a b and C the sides opposite to them, respectively, that is, 
the side a is opposite to A, &c. 

In spherical trigonometry the sines of the angles are propor- 
tional to the sines of the opposite sides. 

sin. A sin. B sin. 



Therefore 34 



sin. a 



sin. b " " sin. c 

Given the three sides abc ; 
Required one of the angles, A. 

sin. (s — b~) sin. (s — c) 



35. 



Sin. 2 i A = 



sin. b sin. c 



16 



232 

Trig. 



ASTRONOMY. 
36. - - Co S .^^= sin -' Ssin -( 5 - a ) 



sin. b sin. c 
In 35 and 36, 2S=a+b+c. 



CHAPTER I. 



ASTRONOMICAL PROBLEMS 



Chap. I. 



A general 
projection for 
connecting 
right ascen- 
sion, decli- 
nation, lon- 
gitude, and 
latitude. 



PROBLEM I. 

Given the right ascension and declination of any heavenly 
body to find its latitude and longitude ; or conversely, given the 
latitude and longitude of a body to find its corresponding right 
ascension and declination. 

From any point as a center 
(Fig. 49) describe a circle Q 
EP<s>, &c. Let this circle 
represent the meridian, which 
passes through the pole of the 
ecliptic E, the pole of the 
v earth's axis P, and through the 
solstitial points 25, and V5>. 
Then the point Aries ( qp) will 
be at the center of the circle 
and V? q 5 25 and Q qp q will be 
lines crossing each other by an angle equal to the obliquity 
of the ecliptic. P p is the celestial meridian, which passes 
through the equinoctial points, and is the first meridian of 
right ascension E °p e is the first meridian of longitude, and, 
of course, the angle E qp P is equal to the obliquity of the 
ecliptic. 
The figure j^t s b e the position of any celestial body, and draw the 

is considered •-,* . i • 7-» ii i • t 

transparent, meridian or right ascension Psp, also draw the meridian of 

and both longitude Es e, draw also qp s. We have now two right-angled 

represented, spherical triangles s D<¥> and t B s, having a common hypo- 

thenuse qp s; the first is the right ascension triangle, the 




PRACTICAL PROBLEMS. 238 

second is the longitude triangle. Let the student observe Chap. i. 
that the line Q q represents a circle, the whole equator ; and 
the point qp represents, in fact, two points, the degree of 
right ascension and the 180th degree. So the point s repre- 
sents two points, and T B is the right ascension from de- 
gree, or from 180 degrees. 

In our figure, the point s is north of both ecliptic and 
equator ; but it might have been between the two, or south of 
both ; hence, to meet every case, the judgment of the opera- 
tor must be called into exercise to perceive a general 
solution. 

Now, having the right ascension and declination of s, we 
find its latitude and longitude thus : 

In the triangle °p Ds, <¥> D and D s are given, and equa- 
tion 32 gives cp s (h); 33 gives the angle s °p D. From 
s<v D subtract B qp D, the obliquity of the ecliptic, and 
there remains the angle s T B.* 

With the angle s °p B, and the side qp s, equation 20 
gives sB the latitude, and 21 gives ^ j B the longitude 

EXAMPLES. 

1. The right ascension of a certain point in the heavens is 
5 h. 7 m. 50 s., or in arc 76° 57' 30" ; and its declination is 
26° 11' 36" N. : 

What is the latitude and longitude of the same point? Fourequa- 

(32.) (33.) tions con - 

TD 76° 57' 30" cos. 9.353454 - - - sin. 9.988651 ^f^nT 
sD 26° 11' 36" cos. 9.952952 - - - cot. 10.308104 

~<Y> "* 78° 19' 3" cos. 9.306406, 26° 47'' 27"cotyio!296766 
BcpJ) - --- 23 27 32 
s^B - - - - 3 19 55 = a 

* In general, take the difference between the angle S Q&M and the 
obliquity of the ecliptic ; and if the angle S HP B is the greater quan- 
tity, the body is north of the ecliptic, otherwise it is south of it. 
When the declination is south, the angle S T D must be added to the 
obliquity of the ecliptic in the first and second quadrants, and sub- 
tracted in the third and fourth. Hence the judgment of the operator 
must be called in to decide the particulars of the case ; or he must 
have a general formula that will give no exercise to the mind. 



234 ASTRONOMY. 

Chap. I . (20.) (21.) 

(h) 78° 19' 3" sin. 9.990911 tan. 10.684611 

(a) 3 19 55 sin. 8.763965 cos. 9.999265 

3° 15' 36" sin. 8.754876 78 18 6 tan. 10.683876 

Thus we determine that the longitude must be 78° 18' 6", 
and the latitude 3° 15' 36" N. 

2. The longitude of the moon, at a certain time, according to 
computation, was 102° 7'; and latitude 5° 14' 15" S. : 

What was the corresponding right ascension and declination ?* 

From these 0* 2 ') ( 33 

examples we opB 77° 53' cos. 9.322019 sin. 9.990215 

might form a g B 50 u , 15 „ cog _ 9- 998 18 3 cot> H.037780 

general rule ; 



batruiesthus cp s 77° 56' 12" cos. 9.320202 5° 21' 27" cot. 11.027995 

formed sel- £cpj) - - 23 27 42 

dom reflect 

principles ; 18 O lO 

n:;:»i: (200 m 

purposes, we (h) 77° 56' 12" sin. 9.990302 tan. 10.670170 

fan back on , fl) 18 Q 15 gin 9.492400 cos. 9.977948 

the primary Ji — — 

equations. 17 41 22 sin. 9.482702 77° 19' 41" tan. 10.648118 

Thus we find that the right ascension distance on the equa- 
tor, from the 180th degree, was 77° 19' 41"; or its right as- 
cension in arc was 102° 40' 19", or in time, 6h. 50m. 41s. 

3. By meridian observations on the moon, at a certain time, 
its right ascension was found to be 16h. 53m. 33s., and its decli- 
nation 17° 51' 36" S. : what was its longitude and latitude? 
Ans. Lon. 254° 9' 14", Lat. 4° 41' 12" N. 

Any num- In the following examples either right ascension and decli- 
ber of the na ti on ma y fo e taken for the data, and the longitude and lati- 

like exam- J • a 

pies can be tude the sought terms, or conversely ; the longitude and 
found. latitude may be the given data, and the right ascension and 

* As the longitude is more than 90° and less than 180°, the moon is 
in the second quadrant of right ascension, and 77° 53' in longitude 
from the equator, and as her latitude is south, it does not correspond 
to B S in the figure, and we give the example to exercise the judg- 
ment of the learner. 



PRACTICAL PROBLEMS, 



235 



declination the required terms. A Nautical Almanac will CHilp - L 
furnish, any number of similar examples. 



R. A. 

h. m. s. 



Dec. 



Lon. 



Lat. 



4 15 47 36 15 58 15 south, 

5 613 22 18 23 2 north, 

6 11 24 44 1 45 28 north, 

7 20 23 33 14 11 9 south, 



238 14 48 4 30 17 north, 

9310 55 5 4 23 south, 

17112 40 152 51 south 

304 47 15 5 2 23 north! 



PROBLEM II. 

Given the latitude of the place, and the declination of the sun 
or star ; to find the semidiurnal arc, or the time the sun or star 
would remain above the horizon; and to find its amplitude, or the 
number of degrees from the east and west points of the horizon, 
where it will rise and set. 

To illustrate this problem we draw Figure 50. Let P Z H, 
&c, represent the celes- 
tial meridian passing 
through the place. Make 
the arc Q Z equal to the 
latitude, then ZP will 
equal the co-latitude. 
The line Hh is every' 
where 90° from Z, and' 
represents the horizon. 
Pp represents the earth's 
axis, and the meridian, 
90° distant from the me- 
ridian of the place ; Q g 
is the equator. From the points Q and q set off d and d ', 
equal to the declination (north or south, as the case maybe) 
and describe the small circle of declination, d Q d ', where this 
circle crosses the circle of the horizon Hh is the point where 
the body ( sun, moon, or star ) will rise or set ( rise on one 
side of the meridian and set on the other, both are repre- 
sented by the same point in the projection ). Through P Q 
p describe the meridian as in the figure, and the right-angled 
spherical triangle P Q C appears ; right angled at R. 




Tables for 
the semidi- 
urnal arc and 
amplitudes 
are computed 
by this prob- 
lem. 

These ex- 
amples do 
not take re- 
fraction into 
account. 



236 ASTRONOMY. 

chapel j n the triangle R Q C, there is given the side R Q, 
the decimation, and the angle opposite R C Q, which is equal 
to the co-latitude. R C, expressed in time, at the rate of 15° 
to one hour, will be the time before and after 6 hours, from 
the time the body is on the meridian to the time it is in the 
horizon ; and the arc C Q is the amplitude. The triangle is 
immediately resolved by equations 26 and 27. 

(27.) Sin. R C = tan. declin. X tan. lat. 

, ao a . „ sin. declin. 

(26. ) Sin. Cq = ; 

v \ w cos. lat. ' 

Observing that the tangent of the latitude is the same as 
the cotangent of the angle R C Q, and the cosine of the lati- 
tude is the same as the sine of R C Q, corresponding to a in 
the equation. 

EXAMPLE. 

The time j n tf w i a tit u d e f 40° N., when the sun's declination is 20° 
examples is, ^ ., what time before and after six will it rise and set, and what 
of course, ap- win })Q Us amplitude? 

parent, be- 
cause it re- (27.) (26.) 
fers directly 2QO ^ 9>561066 gin> 9.534052 
to the sun, 

and not to a 40 tan. 9.923813 cos. 9.884254 

dock. 17° 47' sin. 9.484879 26° 31' sin. 9.649798 

Thus we find that the arc called the ascensional difference, 
is 17° 47', or, in time, lh. 11m. 8s., showing that the sun or 
heavenly body, whatever it may be ( when not affected by 
parallax or refraction), will be found in the horizon 7h. 11m. 
8s. before and after it comes to the meridian. 

Its amplitude for that latitude and declination is 26° 31' 
north of east, or north of west, and, if observed by a compass, 
the apparent deviation would be the variation of the compass. 

2. At London, in Lat. 51° 32' N., the sun's amplitude was 
observed to be 39° 48' toward the north ; what was its declina- 
tion, and what was the apparent time of its rising and setting? 
Ans. Sun's declination, 23° 27' 59" N. 

Sun's rising, 3h. 47m. 32s. ; sun's setting, 8h. 12m. 28s. 



PRACTICAL PROBLEMS. 237 

The amplitude of the sun is frequently observed, at sea, to Chap l 
discover the variation of the compass ; but, by reason of re- 

x ^ Refraction 

fraction, the results are not perfectly accurate. not taken m- 

From the right-angled spherical triangle (Fig. 48) P ZQ, t0 acco * nt > 
we can compute the time when the sun is east or west in po- time that ' the 
sition, and the altitude it must have, when in that position. sun wouId 
The triangle Z is a right angle, P Z is the co-latitude, and "j^" the 
Pq is the co-declination. horizon 

Equation (23) gives the cosine of Z Q, or the sine of the would be in " 

crccisGti 

altitude of the sun when it is east or west — the latitude and while it rose 
declination being given — and equation ( 24 ) will give the in altitude 

k,. o 33' of arc. 

or time irom noon. 

We may also find the altitude and azimuth of the sun, at 
6 o'clock, by making use of a triangle, formed by drawing a 
vertical through Z s JV; C S, the given declination, will be its 
hypothenuse, and P Ch, the latitude, will be the arc of its 
angles. 

By means of right-angled spherical trigonometry, as com- 
prised in the equations from 20 to 33, we can resolve all pos- 
sible problems that can occur in astronomy, pertaining to the 
sphere ; but, for the sake of brevity, mathematicians, in some 
cases, use oblique-angled spherical trigonometry, which is 
nothing more than right-angled trigonometry combined and 
condensed. 

PROBLEM III. 

Given, the latitude of the place of observation, the sun's de- The sun's 
clination, and its altitude above the horizon, to find its meridian dlstance 

from the me- 

distance, or the time from apparent noon. ridian, as 

There is no problem more important in astronomy than me asured 
that of time. No astronomer puts implicit faith in any chro- as a ce -^* 
nometer or clock, however good and faithful it may have and on the 
been ; and even to suppose that a chronometer runs true, it ^j^ ab s 
can only show time corresponding to some particular me- ence, is the 
ridian ; and hence, to obtain local time, we must have some raeasure of 

. time from aD- 

method, directly or indirectly, of finding the sun s distance parent noon 
from the meridian. 

When the center of the sun is on any meridian, it is then 
and there apparent noon ; and the equation of time will be the 



238 



ASTRONOMY 



Chap. i. interval to or from mean noon; but none, save an astronomer 

Great im _ in an observatory, can define the instant when the sun is on 

portance of the meridian ; no one else has a meridian line sufficiently defi- 

is pro em. ^.^ a ^ accurate, and with him it is the result of great care, 

combined with a multitude of nice observations. 

To define the time, then (when anything like accuracy is 
required ), we must resort to observations on the sun's al- 
titude. 

It is evident that the altitude of the sun is greater and 
greater from sunrise to noon, and from noon to sunset the al- 



Direct me- 
ridian obser- 
vations not 



curate. 



Proper times 
of observa- 
tion. 



generally ac titude is continually becoming less. If we could determine, 
by observation, exactly when the sun had the greatest alti- 
tude, that moment would be apparent noon ; but there is a 
considerable interval, some minutes, before and after noon, that 
it is difficult to determine, without the nicest observations, 
whether the sun is rising or falling ; therefore, meridian ob- 
servations are not the most proper to determine the time. 

From two to four hours before and after noon ( depending 
in some respects on the latitude ), the sun rises and falls most 
rapidly ; and, of course, that must be the best time to fix 
upon some definite instant ; for every minute and second of 
altitude has its corresponding time from noon ; and thus the 

time and altitude have 
Fig. 51. . ._ 

a scientific connection, 

which can only be disen- 
tangled by spherical tri- 
gonometry. But we 
proceed to the problem. 
Draw a circle, P Z 
Q M, &c, (Fig. 51), 
^representing the meri- 
dian; Z is the zenith, 
and Z JV is the prime 
vertical ; Hh is the ho- 
rizon; Z Q is an arc 
equal to the given lati- 
tude ; Q q is the equa- 
tor, and, at right angles to it, we have the earth's axis, P S. 



Description 
of the figure. 




PRACTICAL PROBLEMS. 239 

Take H a, h a, equal to the observed altitude of the sun, chap, l 
and draw the small circle, a a, parallel to the horizon, Hh. 
From the equator take Q d, qd, equal to the decimation of 
the sun, and draw the small circle, d d, parallel to Q q. 
Where these two small circles, a a, dd, intersect, is the posi- 
tion of the sun at the time. 

From Z draw the vertical, Z Q JV, and from P draw the 
meridian through the sun, P Q S. The triangle, P Z Q, 
has all its sides given, from which the angle, ZP Q, can be 
computed; which angle, changed into time at the rate of 15° 
to one hour, will give the time from noon, when the altitude 
was taken. If the time, per watch, should agree with the 
time thus computed, the watch is right, and as much as it 
differs is the error of the watch. 

The side, Z Q, is the complement of the altitude, P Q The obser - 
is the complement of the declination, and P Z is the comple- fines and 
ment of the latitude, and equation ( 35 ) or ( 36 ), will solve points out a 
the problem ; that is, find the angle, P, which can be made tnang e ' 
to correspond to A, in the equation. But, in place of using 
the complement of the latitude, we may use the latitude it- 
self; and, in place of using the complement of the altitude, 
we may use the altitude itself; provided we take the cosine, 
when the side of the triangle calls for the sine ; for it would 
be the same thing. By thus taking advantage of every cir- 
cumstance, ingenious mathematicians have found a less 
troublesome practical formula than either (35) or (36) would Mathema- 
be ; but we cannot stop to explain the modifications and tlcians make 

1 L great exer 

changes in a work like this; we contemplate doing so in tions to ab- 
a work more appropriate to such a purpose ; the student must breviate 

. , . . practial ope- 

be content with the following practical rule, to find the time rations. 
of day, from the observed altitude of the sun, together ivith its 
polar distance, and the latitude of the observer. 

Rule 1. — Add together the altitude, latitude, and polar dis- Practical 
tance, and divide the sum by two. From this half sum subtract 
the altitude, reserving the remainder. 

2. — Take the arithmetical complement of the cosine of the lati- 
tude, the arithmetical complement of the sine of the polar distance, 
the cosine of the half sum, and the sine of the remainder. Add 




240 ASTRONOMY. 

Chap - t - these four logarithms together, and divide the sum fo} two; the 
result is the logarithmetic sine of half the hourly angle. 

3.— This angle, taken from the Tables, and converted into 
time at the rate of four minutes to one degree, will be the 
time from apparent noon ; the equation of time applied, will 
give the mean time when the observation was made * 

* The instrument for taking alti- 
rant andlex- .^^X^ tudes at sea, or wherever the observer 

tant and re- ^JJf // \l\@j) ma y na ppen to be, is a quadrant or 

sextant, according to the number of 
?.i!v the same / Jr-^rWf*^ degrees of the arc. It is made on the 

principle of reflecting the image of one 
body to another, by means of a small 
mirror revolving on a center of motion, 
carrying an index with it over the arch. Nearly opposite 
to the index mirror is another mirror, one half silvered, the 
other half transparent, called the horizon glass. Directly op- 
posite to the horizon glass is the line of sight, in which line 
there is sometimes placed a small telescope. The line of 
sight must be 'parallel to the plane of the instrument. The 
two mirrors must be perpendicular to the plane of the instru- 
ment. To be in adjustment, the two mirrors, namely the in- 
dex glass and horizon glass, must be parallel, when the index 
stands at 0. 

To examine whether a sextant is in adjustment or not, 
proceed as follows : 

1. Is the index mirror perpendicular to the plane of the in- 
strument ? 

Put the index in about the middle of the arch, and look 
into the index mirror, and you will see part of the arch re- 
flected, and the same part direct; and if the arch appears 
perfect, the mirror is in adjustment; but if the arch appears 
broken, the mirror is not in adjustment, and must be put so 
by a screw behind it, adapted to this purpose. 

2. Are the mirrors parallel when the index is at 0? 
Place the index at 0, and clamp it fast, then look at some 

well-defined, distant object, like an even portion of the dis- 



PRACTICAL PROBLEMS 



241 



EXAMPLE. 

In latitude 39° 46', north, when the sun's declination was 
3° 27', north, the altitude of the sun's center, corrected for 
refraction, index error, &c, was 32° 20', rising ; what was 
the apparent time ? 



Chap. I. 



Altitude, 
Latitude, 
Polar dis., 



32 
39 

86 



20 
46 
33 



cos. comple. 
sine comple. 



2)158 39 



79 
32 



19 

20 



30 cosine 



46 59 30 sine 



\ZP O 24 50 



30 sine 

2 



- .114268 

- .000788 

9 .267652 

9 .8 64090 

2)19 .246798 

~9~623399 



The hourly angle is 49 41 0. which, converted into time, 
gives 3h. 18 m. 44 s., the time from apparent noon, and, as 

tant horizon, and see part of it in the mirror of the horizon 
glass, and the other part through the transparent part of the 
glass ; and, if the whole has a natural appearance, the same 
as without the instrument, the mirrors are parallel; but, if 
the object appears broken and distorted, the mirrors are not 
parallel, and must be made so, by means of the lever and 
screws attached to the horizon glass. 

3. Is the horizon glass perpendicular to the plane of the in- 
strument ? 

The former adjustments being made, place the index at 0, 
and clamp it ; look at some smooth line of the distant horizon, 
while holding the instrument perpendicular ; a continued, un- 
broken line will be seen in both parts of the horizon glass ; 
and if, on turning the instrument from the perpendicular, the 
horizontal line continues unbroken, the horizon glass is in full 
adjustment ; but, if a break in the line is observed, the glass 
is not perpendicular to the plane of the instrument, and must 
be made so, by the screw adapted to that purpose. 

After an instrument has been examined according to these 
16 u 



242 ASTRONOMY. 

chap. i. the sun was rising, it was before noon, and the apparent time 

was 8 h. 41 m. 16 s. 
An arc may j^ 0( j observer, with a good instrument, in favorable cir- 

be measured ° ° 

by the quad- cumstances, can define the time, from the sun's altitude, to 

rant within w ithin three or four seconds. 

An artificial At sea ' tne observer brings the reflected image of the sun 

horizon. to the horizon, and allows for the dip (Tables p.25). On shore, 
where no natural horizon can be depended upon, resort is had 
to an artificial horizon, which is commonly a little mercury 
turned out into a shallow vessel, and protected from the wind 
by a glass roof. The sun, or any other object, may be seen 
reflected from the surface of the mercury ( which, of course, 
is horizontal ), and the image, thus reflected, appears as much 
below the natural horizon as the real object is above the hori- 
zon ; and, therefore, if we measure, by the instrument, the 
angle between the object and its image in the artificial hori- 
zon, that angle will be double the altitude. 

When mercury is not at hand, a plate of molasses will do 
very well ; and in still, calm weather, any little standing pool 
of water may be used for an artificial horizon. 

Observations taken in an artificial horizon are not affected 
by dip, but they must be corrected for refraction and index 
error, and, if the two limbs of the sun are brought together, 
its semidiameter must be added after dividing by two. 
a practical The following example is from a sailor's note book : 

example. u Qr ^ jg^ Qf ^ jg^ ^ ^ ^ fa^fa 350 2 T, 

north, longitude, 54° 10', west, by account, at 7 h. 43 m., per 
watch; the altitude of the sun's lower limb was 32° 51', ris- 
ing; the hight of the eye was eighteen feet, and the index 

directions, it may be considered as in an approximate adjust- 
ment — a re- examination will render it more perfect — and, 
finally, we may find its index error as follows : — measure the 
sun's diameter both on and off the arch — that is, both ways 
from 0, and if it measures the same, there is no index error ; 
but if there is a difference, half that difference will be the in- 
dex error, additive, if the greatest measure is off the arch, 
sub tractive, if on the arch. 



PRACTICAL PROBLEMS. 243 

error of the sextant was 2' 12" additive. What was the er- Ch *p. i. 
ror of the watch?" 

PREPARATION. 

Time, per watch, - - - 7 h. 43 m., morning. Preparations 

Longitude, 54° 10', in time, 3 38 to be made 

' according to 

Estimated mean time at Greenwich, 11 h. 21 m. ch-cnm- 

The declination of the sun at mean noon, Greenwich time, stances - 
was 19° 38' 29", increasing, the daily variation being 13' ; 
the variation, therefore, for 39', the time before noon, was 
21", subtractive. Hence, the declination of the sun, at the 
time of observation, was 19° 38' 8", north, and the polar dis- 
tance, 70° 21' 52". 

Observed altitude, - - - - 32° 51' 00" 

Index error, ----- -|- 2 12 

Semidiameter, - - - - - -(-15 49 

Refraction, ------ — 128 

Dip of the horizon, - - - - — 413 

True altitude of sun's center, - - 33° 3' 20" 

Altitude, 33° 3' 20" 

Latitude, 36 21 cos. complement, .093982 

Polar dis. , 70 21 52 sin. complement, .026013 
2 )139 46 12 

69 53 6 cosine, - - 9.536470 
33 3 20 

36~~4~9~46 sine, - - 9.777770 

2)19.434235 

\ hourly angle, 31 25 30 sine, - - 9.717117 

This angle corresponds to 4h. 11m. 24 s., or, in reference 
to the forenoon, 7 h. 48 m. 36 s., apparent time. 

On the 18th of May, noon, Greenwich time, the equation B ? obser " 
of time was 3m. 54s., subtractive; therefore, the true mean taken at dif< , 
time, at ship, was - - - 7 h. 44 m. 42 s. ferent times 

Time, per watch, 7 43 " the sa ™ e 

- 1 place, the 

Watch slow, ... 1 42 rate of the 

. . watch can he 

A short time before this observation was taken, the watch determined. 



h. m. s. 




7 43 00 




3 56 39 




11 39 39 




19 12 




11 20 27 




7 44 42 




3 35 45= 


=53° 56' west. 



244 ASTRONOMY. 

Chap. i. W as compared with a chronometer in the cabin, which was 
too fast for mean Greenwich time, 19 m. 12.5 s., according to 
estimation from its rate of motion. The chronometer was 
fast of watch by 3 h. 56 m. 39 s. What was the longitude of 
the ship? 

Time of observation, per watch, 
Diff. between watch and chron., 
Time, per ch., at observation, 
Chron. fast of Greenwich time, 

Greenwich mean time, 
Mean time at ship, 

Longitude in time, 

How to de- The longitude is west, because it is later in the day, at 
ci e rom e Gre^^ei* than a £ the ship. This example explains all the 

observations i i i 

whether the details of finding the longitude by a chronometer. 

longitude is -g taking advantage of the observations for time on shore. 

east or west. J ° ? . . . 

How to de- we may draw a meridian line with considerable exactness ; 
termine and f or instance, in the last observation (if it had been on land), 
meridian i Q ^h- H m « 24 s., after the observation was taken, the sun 
line. would be exactly on the meridian ; and if the watch could be 

depended upon to measure that interval with tolerable accu- 
racy, the direction from any point toward the sun's center, 
at the end of that interval, would be a meridian line. Sev- 
eral such meridians, drawn from the same point, from time to 
time, and the mean of them taken, will give as true a me- 
ridian as it is practical to find ; although, for such a purpose, 
a prominent fixed star would be better than the sun. 
Absolute The problem of time includes that of longitude, and find- 
ing the difference of longitude between two places always re- 
solves itself into the comparison of the local times, at the same 
instant of absolute time. When any definite thing occurs, 
wherever it may be, that is absolute time. For instance, 
the explosion of a cannon is at a certain instant of absolute 
time, wherever the cannon may be, or whoever may note the 
event ; but if the instant of its occurrence could be known 
at far distant places, the local clocks would mark very diffe- 



t:me. 



PRACTICAL PROBLEMS. 245 

rent hours and- minutes of time; but such difference would be chap, l 
occasioned entirely by difference of longitude; the event is 
the same for all places — it is a, point in absolute time. 

Thus any single event marks a point in absolute time. If Absoiuii 
the same event is observed from different localities the diffe- . mean3 of 
rence in local time will give the difference in longitude. But events. 
a perfect clock is a noter of events, it marks the event 

■*■ J a noter oJ 

of noon, the event of sunrise, the event of one hour after events, when 

noon, &c. ; and if we could have perfect confidence in this ll rmis trae ' 

x m but not other- 

marker of events, nothing more would be necessary to give us W i se . 

the local time at a distant place. The time, at the place 
where we are, can be determined by the altitude of the sun, 
or a star, as we have just seen. But, unfortunately, we can- 
not have perfect confidence in any chronometer or clock ; and 
therefore we must look for some event that distant observers 
can see at the same time. 

The passage of the moon into the earth's shadow is such Eclipses are 
an event, but it occurs so seldom as to amount to no practical ^^' mark 
value. The eclipses of Jupiter's satellites are such events, absolute 
but they cannot be observed without a telescope of consider- time ' but for 

^ A common pur- 

able power, and a large telescope cannot be used at sea. p0 ses they 

Hence these events are serviceable to the local astronomer are of httIe 

value. 

only ; the sailor and the practical traveler can be little bene- 
fited by them. The moon has comparatively a rapid motion 
among the stars ( about 13° in a day), and when it comes to 
any definite distance to or from any particular star, that cir- 
cumstance may be called an event, and it is an event that can 
be observed from half the globe at once. 

Thus, if we observe that the moon is 30° from a particular The raotion 

. of the moon 

star, that event must correspond to some instant ot absotute among t i. e 
time ; and if we are sufficiently acquainted with the moon, stars, may be 
and its motion, so as to know exactly how far it will be from ;, • „ e - -. 

* w 3.S 3.11 lriUcX 

certain definite points ( stars ) at the times, when it is noon, moving 
3, 6, 9, &c, hours at Greenwich, then, if we observe these roui ! d a circ ' e 

' marking ab- 

events from any other meridian, we thereby know the Green- solute time, 
wich time, and, of course, our longitude. 

Finding the Greenwich time by means of the moon's angu- 
lar distance from the sun or stars, is called taking a lunar; 



246 ASTRONOMY. 

Chap. i. and it is probably the only reliable method for long voyages 
at sea. 

If the motion of our moon had been very slow, or if the 
earth had not been blessed with a moon, then the only 
methods, for sea purposes, would have been chronometers and 
dead reckoning. For a practical illustration of the theory of 
lunars, we mention the following facts. 

Lunar ob- j Q a gea j ourna ] f 1823, it is stated that the distance of 

servations n- " 

lustrated by the moon from the star Antares was found to be 66° 37' 8", 
an example. w j ien tfr e observation was properly reduced to the center of the 
earth, and the time of observation at ship was September 
16th, at 7h. 24m. 44s., p. m., apparent time. 

By comparing this with the Nautical Almanac, it was 
found that at 9, p. m., apparent time at Greenwich, the dis- 
tance between the moon and Antares was 66° 5' 2", and at 
midnight it was 67° 35' 31"; but the observed distance was 
between these two distances, therefore the Greenwich time 
was between 9 and 12, p. m., and the time must fall between 
9 and 12 hours, in the same proportion as 66° 37' 8" falls 
between the distances in the Nautical Almanac; and thus an 
observer, with a good instrument, can at any moment deter- 
mine the Greenwich time, whenever the moon and stars are 
in full view before him. ^ 

The moon, in connection with the stars in the heavens, 
may be considered a public clock ( quite an enlargement of 
the town- clock ), by which certain persons, who understand 
the dial plate and the motion of the index, and who have the 
necessary instrument, can read the Greenwich time, or the 
time corresponding to any other meridian to which the com- 
putations may be adapted, 
observed The angular distances from the moon to the sun, stars, 
distances,^ an( j pi anetgj ag p ut <j own j n the Nautical Almanac, eorre- 

tances as sponding to every third hour, are distances as seen from the 
peen from cen £ er f the earth, and when observations are taken on the 

iH© C6nt6r of" 

ti.e earth, surface the distance is a little different ; the position of the 

moon is affected by parallax and refraction, the sun or star 

i the % refraction alone ; and therefore a reduction is necessary, 

distance. which is called clearing the distance. This is done by spheri- 



PROPORTIONAL LOGARITHMS. 247 

cal trigonometry. The distance between the moon and star Chap - l 
is observed, the altitudes of the two bodies are also observed. 
The co-altitudes come to the zenith, and the co-altitudes, 
with the distance, form three sides of a spherical triangle, 
from which the angle at the zenith can be computed. Then 
correct the altitude of the moon, for parallax and refraction, 
and the star for refraction, and find the true altitudes and co- 
altitudes, and the true co- altitudes and angle at the zenith 
give two sides, and the included angle of a spherical triangle, 
and the third side, computed, is the true distance. 

An immense amount of labor has been expended by mathe- 
maticians, to bring in artifices to abbreviate the computation 
of clearing lunar distances ; and they have been in a measure 
successful, and many special rules have been given, but they 
would be out of place in a work of this kind. 

PROPORTIONAL LOGARITHMS. 

In every part of practical astronomy there are many pro- Proportional 
portional problems to be resolved, and as the terms are lo g arith m s — 

i ' . 7i« -i-ii t • an explana- 

mostly incommensurable, it would be very tedious, in most tion of the 
cases, to proceed arithmetically, we therefore resort to loga- construction 
rithms, and to a prepared scale of logarithms, very appropri- CTiven 
ately called proportional logarithms. 

The tables of proportional logarithms commonly correspond 
to one hour of time, or 60' of arc, or to three hours of time. 
The table in this book corresponds to one hour of time, or 
3600 seconds of either time or arc. To explain the construc- 
tion and use of a table of proportional logarithms, we propose 
the following problem : 

At a certain time, the moon's hourly motion in longitude was 
33' 17" ; how much would it change its longitude in 13m. 23s. ? 

Put x to represent the required result, then we have the 
following proportion : 

m. m. s. 



60 : 


13 23 : 


: 33 17 : x; 


3600 : 


: 13 23 : 


: 33 17 : x. 



Or 

Divide the first and second terms of this proportion by the 
17 



248 ASTRONOMY. 

Chap - !• second, and the third and fourth by the third, then we have 

3600 x 



13.23 33.17 

Divide the third and fourth terms by x, and multiply the 
same terms by 3600, and the proportion becomes 

3600 , 3600 3600 

: 1 : : 



13.23 x * 33.17' 

Multiplying extremes and means, using logarithms, and re- 
membering that the addition of logarithms performs multipli- 
cation, 

T , '■ . 3600 /3600\ /3600\ 

Then we have log. — ^ = log. (^^-J +log. (^-j. 

By the construction of the table, the proportional logarithm 

of 1" is the common logarithm of — - — ; of 2" is the com- 

i -n. * 3600 * w • 3600 s * 3600 
mon logarithm of — ^ — ; 01 o is — 5 — , &c, to ^7^; 

hence the proportional logarithm of 3600 is 0. 
We now work the problem : 

13 23 - - - P. L. 6516 

33 17 - - - P. L. 2559 

Result, - - 7 25i - - - P. L. 9075 

Examples EXAMPLES FOR PRACTICE. 

given to il- . m 

instrate the 1. When the sun's hourly motion in longitude is 2' 29", 
practical nti- w k at jg ft g c }i an g e f longitude in 37 m. 12 s.? 

lityofpropor- 

tional logar- IS ' ns - x °^ • °' 

ithm?. 2. When the moon's declination changes 57".2 in one hour, 

what will it change in 17 m. 31 s. ? Ans. 16".8. 

3. When the moon changes longitude 27' 31" in an hour, 
how much will it change in 7 m. 19 s. ? Ans. 3' 21". 

4. When the moon changes her right ascension lm. 58 s. 
in one hour, how much will it change in 13 m. 7 s.? 

Ans. 25".8. 



PROPORTIONAL LOGARITHMS 



249 



N. B. This table of proportional logarithms will solve any C hap. i. 
proportion, provided the first term is 60, or 3600 ; therefore, 
when the first term is not 60, reduce it to 60, and then use 
the table. 



EXAMPLES. 

1. If the sun's declination changes 16' 33" in twenty-four 
hours, what will be the change in 14 h. 18 m. ? 



Statement, 

Or, 

Or, 



24 
12 
60 



14.18 

7.09 

35.45 



16' 33" 

16' 33" 

16' 33" 
35' 45" 



Examples 
given to il- 
lustrate the 
practical uti- 
lity of propor- 
tional logar 
ithms, 



P. L. 
P. L. 



5594 
2249 



Ans. 9' 51".5 P. L. 7843 



2. If the moon changes her declination 1° 31' in twelve 
hours, what will be the change in 7 h. 42 m. ? Ans. 58'. 

Conceive degrees and minutes to be minutes and seconds, 
and hours and minutes to be minutes and seconds. 

When 60 minutes or 3600 seconds are not the first term of 
a proportion, the result is found by taking the difference of 
the proportional logarithms of the other term for the P. L. 
of the sought term, as in the following example : 

The moon's hourly motion from the sun is 26' 30'', what 
time will it require to gain 30" ? 



Statement, 26' 30" : 60m. : 30' 



30" 
60 m. 



x 

P. L. 
P. L. 



Other ex 

amples. 



2.0792 
0.0000 



Product of extremes, 
Result, 



2.0792 
26' 30" P. L. sub. 3549 

lm. 77 p7l~ 1.7243 



3. The equation of time for noon, Greenwich, on a certain 
day, was 6 m. 21 s. ; the next day, at noon, it was 6 m. 43 s. : 
what was it corresponding to 3 h. 42 m., p. m., in longitude 
74° west, on the same day ? Ans. 6 m. 29 s. 



250 ASTRONOMY. 

CHAPTER II. 

GENERAL PROBLEM. 

Chap. ii. Given, the motions of sun and moon, to determine their appa- 
a o-enerai ren t positions at any given time ; from which results their appa- 
probiem pre- ren t relative situations, and the eclipses of the sun and moon. 
fi« com nta° "^ s problem covers two-thirds of the science of astronomy, 
tionsofeciip- and includes many minor problems ; therefore a brief or hasty 
ses * solution must not be expected. 

From the foregoing portions of this work, the reader is 
supposed to have acquired a good general knowledge of the 
solar and lunar motions, and the tables give all the particu- 
lars of such motions ; and all the artifices and ingenuity that 
astronomers could devise, have been employed in forming and 
arranging these tables, for the double purpose of facilitating 
the computations and giving accuracy to the results. 

The tables, in general, must be left to explain themselves, 
and the mere heading, combined with the good judgment of 
the reader, will furnish sufficient explanation, in most in- 
stances ; but some of them require special mention. All the 
tables are adapted to mean time at Greenwich. 

EXPLANATION OF TABLES. 

a very ge- Table IV contains the sun's mean longitude, the longi- 
nerai and ^^ Q Q ^ ^ g p er ip. ee /^gh diminished by 2°), and the Argu- 

comprehen- 1 c \ «/ /> «? 

sive expiana- ments * for some of the small inequalities of the sun's appa- 

tion of the rent mot j on _ 
tables. 



Explanation * The term, argument, in astronomy, means nothing more than a 
of the term correspondence in quantities. Thus, each and every degree of the 
argument. sun's longitude corresponds with a particular amount of declination ; 
and therefore, a table could be formed for the declination, and the ar- 
gument would be the sun's longitude. 

The moon's horizontal parallax and semidiameter vary together, 
and each minute of parallax corresponds to a particular amount of se- 
midiameter; hence, a table can be made for finding the semidiameter, 
and the arguments would be the horizontal parallax. But the hori- 



EXPLANATION OF TABLES. 251 

Argument I; corresponds to the action of the moon; Ar- Chap, ii, 
gument II, to the action of Jupiter; Argument III, to Ve- 
nus ; and Argument N, is for the equation of the equinoxes, 
and corresponds with the position of the moon's node ; and, 
by inspecting the column in the table, it will be perceived 
that the argument runs round the circle in a little more than 
eighteen years, as it should; and thus, by inspection, we can 
obtain an insight as to the period of any argument in the 
solar or lunar tables. 

The object of diminishing the mean longitude and perigee Explanation 
of the sun by 2°, is to render the equation of the center al- of the so ' ar 

... . tables. 

ways additive ; for if 2° are taken from the longitude, and 2° 
added to the equation of the center, the combination of the 
two quantities will be the same as before ; and, as the equa- 
tion of the center is always less than 2°, therefore, 2° added 
to its greatest minus value, will give a positive result. By 
the same artifice all equations may be rendered always posi- 
tive. The 2°, taken from the mean longitude, are restored by 
adding 1° 59' 30" to the equation of the center, and 10" to 
each of the other equations ; hence, to find the real equation 
of the center corresponding to any degree of the anomaly, 
subtract 1° 59' 3" from the quantity found in the table. 

Table XII, shows the time of the mean new moon, &c, 
in January, diminished by fifteen hours, to render the correc- 
tions always additive, The fifteen hours are restored by add- 
ing 4h. 20 m. to the first equation, 10 h. 10 m. to the second, 
10 m. to the third, and 20 m. to the fourth. 

Argument I, corrects for the action of the sun on the lunar 

zontal parallax and semidiameter of the moon depend (not solely) on the 
moon's distance from its perigee; hence, a table can be formed giving 
both horizontal parallax and semidiameter; which arguments are the 
anomaly. In other words, an argument may be called an index, and 
when the arguments correspond to points in a circle, or to the differ- 
ence of points in a circle, the circle may be considered as divided into 
1000 or 100 parts, then 500, or 50, as the case may be, would corre- 
spond to half a circle, and so on in proportion. This mode of dividing 
the circle has been adopted, with certain limitations, to avoid the 
greater labor of computing by denominate numbers. 



252 ASTRONOMY. 

Chap. ii. orbit ; Argument II, corrects for the mean eccentricity of the 
lunar orbit ; Argument III, corrects for the different combina- 
tions of the solar and lunar perigee ; and Argument IV, cor- 
rects for the variation occasioned by the inclination of the 
lunar orbit to the ecliptic ; N. shows the distance from or to 
the nodes. 
Tables ad- New and full moons, calculated by these tables, can be de- 

Tnodicti 1 6 P en ^ e< l u P on within four minutes, and commonly much nearer; 

motion of the but when great accuracy is required, the more circuitous and 

moon, by e i a jj rate method of computing the longitudes of both sun 

which new V 

and full and moon must be employed. 

moons can be Tables XIII, XIV, and XV, are used in connection with 

computed. TaWeXIL 

Explanation Table XVI, shows the reduction of the latitude, and also of 
table Un * ne moon ' s horizontal parallax, corresponding to the latitude, 
occasioned by the peculiar shape of the earth, and the dimi- 
nution of its diameter as we approach the poles. The table 
is put in this place because of the convenient space in the page. 

Table XVII, and the following tables to No. XXX, contain 

the arguments and epochs of the moon's mean longitude, evee- 

tion, &c, necessary in computing the moon's true place in 

the heavens. 

The method The argument for evection is diminished by 29' ; the ano- 

theTme^on? ma ly by 1° 59 ', the variation by 8° 59', and the longitude 

gitude of the by 9° 44', and the balances are restored by adding the same 

amounts to the various equations, which, at the same time, 

renders the equation affirmative, as explained in the solar 

tables. 

The arguments in Table xxxn, are also arguments for polar 
distance, or latitude, in Table xxviii. Anything like a minute 
explanation of, these tables would lead us too far, and not 
comport with the design of this work. The use of the tables 
will be shown by the examples. 

We have carried the mean motions of the sun and moon 
only to five minutes of time — and this is sufficient for all 
practical purposes — for we can proportion to any interme- 
diate minute or second, by means of the hourly motions. 



PRACTICAL PROBLEMS. 253 

Chap. II. 
PROBLEM I. — 

From the solar tables find the sun's longitude, hourly motion 
in longitude, declination, semidiameter and equation of time; 
and for a specific example, find these elements corresponding to 
mean time, at Greenwich, 1854, May 26 d. 8 h. 40 m. 

To find the sun's declination, spherical trigonometry gives 
us the following proportion : (Eq. 20, page 231.) 

As radius 10.000000 

Is to sin. of O's Ion. (65° 12' 15") - - 9.957994 
So is sin. of obliq. of the eclip. ( 23° 27' 32") 9.599900 
To sin. declination N., 21° 10' 54" - - 9.557894 

In nearly all astronomical problems, time is reckoned from 
noon to noon — from hour to 24 hours. 

When the given time is apparent, reduce it to mean time, 
and when not at Greenwich, reduce it to Greenwich time, by 
applying the longitude in time. — ( This is necessary because 
the tables are adapted to Greenwich mean time. \ 

From Table IV, and opposite the given year, take out the 
whole horizontal line of numbers ( headed as in the table ) 
and from Tables V, VII, VIII, take out the numbers corre- 
sponding to the month — day of the month — hour and 
minute of the day, as in the following example. 

Add up the perpendicular columns, as in compound num- The sun's 
bers, meeting entire circles in everv column, and the sums or dlstances 

' J ° * ' from its peri- 

surplus, as the case may be, will give the mean values of all gee point is 
the quantities for the given instant. called its 

Subtract the longitude of the perigee from the mean Ion- maly# 
gitude, and the remainder will be the mean anomaly ; which is 
the argument for the equation of the center. 

With the respective arguments take out the corresponding 
equations, all of which add to the mean longitude, and the 
true longitude of the sun from the mean equinox will be found. 

With the argument N * take out the equation of the equi- 

* The reason why N is not applied with the other equations is be- 
cause it is sometimes negative. 



254 



ASTRONOMY. 



Chap - u - noxes from Table X, and apply it according to its sign, and 
the result will be the true longitude from the true equinox. 







M. Lon. 


Lon. Perig. ] 


I. 


II. 


III. 


N.j 


S. o ' " 


S. O ' " 




1854 




9 8 48 48 


9 8 25 29 


073 


998 


902 


809 




May 


3 28 16 40 


20 


59 


301 


206 


18 




26 d 
8h 


24 38 28 
19 43 


4 


844 
11 


63 



43 



4 







40m 


139 




987 


362 


151 


831 




2 2 5 18 


9 8 25 53 


Eq. of center 3 6 42 


2 2 518 


I 10 

II 13 


4 23 39 25 = Mean anomaly. 


III 8 




2 5 12 31 




Eq. of the equinox — 16 


Sun's hourly motion in lon. 2' 24" 


True lc 


>n. 


2 5 12 15 


" semidiameter, 


15' 49' 



These prin- To find the equation of time to great accuracy. 



ciples were 
explained on 
pages 94 
and 95. 



O t o 

63 16 10 
65 12 15 



By equation 21, page 231, we find 
the sun's R. A., - 

Subtract this from the sun's lon., - 

Equatorial point is west of mean east- 
ward motion by 

From the equation of the center, as 

just found, 
Subtract the constant of the table, 

The sun east of its mean place, 
Subtract ( b ) from ( a ) because one 

is east, the other west, and we 

have the arc - - 48' 53" 

This arc, converted into time, gives 3 m. 15.5 s. for the 
equation of time at this instant, and the sun will not come to 
the meridian at mean noon, but 3 m. 15| s. afterward 
Hence, to convert mean into apparent time, in the month of 
May, add the equation of time. 



1° 56' 5" {a t 



3 6 42 
1 59 30 

1 7 12 (b) 



PRACTICAL PROBLEMS. 255 

Thus, in general, we can determine the exact amount of Chap. ii. 
the equation of time, by means of the two arcs ( a ) and ( b ) 
( which are roughly tabulated on page 95 ), and, without 
strictly scrutinizing each particular case, we can determine 
whether we are to take the sum or difference of the arcs by 
inspecting the table on page 95, or by referring our results to 
some respectable calendar. 

EXAM PLE. 

2. What will be the sun's longitude, declination, right as- 
cension, hourly motion in longitude, semidiameter of the sun, 
and equation of time corresponding to 20 minutes past 9, 
mean time at Albany, N. Y., on the 17th of July, 1860 ? 

N. B. At this time the sun will be eclipsed. 

Ans. Lon. 214° 38' 21"; Dec. 21° 12' 48". 

E. A., in time, 7h. 46m. 15s. ; Eq. of time to add to apparent 
time, 5m. 46.2s.; hourly motion in lon, 2' 23"; S. D, 15' 45.6". 

PROBLEM II. 

From Tables XI, XII, and XIII, to find the approximate time 
of new and full moons. 

Take the time of new moon, and its arguments, from Table 
XI, corresponding to January of the given year, and take 
as many lunations, from the following table, as correspond to 
the number of the months after January, for which the new 
moon is required; add the sums, rejecting the sums corre- 
sponding to whole circles, in the arguments, and in the column 
of days, rejecting the number corresponding to the expired 
months, as indicated by Table XIII; the sums will be the 
mean new moon and arguments for the required month. 

When a full moon is required, add or subtract half a luna- Add the 
tion. Sometimes one more lunation than the number of the numberoflu - 

. m nations ne- 

month after January, will be required to bring the time to cessary to 
the required month, as it occasionally happens that two luna- hrin s the re - 

. . , ., suit to the re- 

tions occur m the same month. quired time 

Apply the equations corresponding to the different argu- of year. 
ments taken from Table XIV, and their sum, added to the 
mean time of new or full moon, will give the true mean time 
of new or full moon for the meridian of Greenwich^ within 
four minutes, and generally within two minutes. 



256 



ASTRONOMY. 



Chap. ii. For the time at any other meridian apply the time corre- 
sponding to the longitude. 

EXAMPLES. 

1. Required the approximate time of new moon, in May, 
1854, corresponding to the day of the month, and the time of 
the day, at Greenwich, England, Boston, Mass., and Cincin- 
nati, Ohio. 



January. 


Mean 


N. 


Moon. 


I. 


II. I III. 


IV. j N. 


1854, 
Four Luna. 

Table XIII. 


27d. 
118 


18h 
2 


. 14m. ! 
56 


0761 
3234 


1168 | 19 
2869 61 


04 1 668 
96 j 341 


145 
120 


21 


10 


3995 


4037 1 80 | 00 | 009 


May, 

I. 
II. 
III. 
IV. 


25 


21 

6 
4 


10 
46 
14 
17 

20 


N shows an eclipse of the 
sun — visible in the United 
States. 


May, 


26 


8 


47 






_ 



8 h. 47 m., p. m. 
4 44 



New nD mean time at Greenwich, 
Boston, Longitude, 

New #) Boston time, 

Cincinnati, Longitude from Boston, 

New <§) Cincinnati time, 

2. Required the approximate time of full moon, in July, 
1852, for the meridian of Greenwich, and for Albany time, 
New York. 



4 3 
53 

3 10 



January. 


Mean N. 


Moon. 


I. 


II. 


III. 


IV. 


N. 
538 


1852, 


20d 


llh 


.53m. 


0549 


3239 


38 


27 


Five Luna. 


147 


15 


40 


4042 


3586 


76 


95 


426 


Half Luna. 


14 


18 


22 


404 


5359 

2184 


58 

72 


50 

72 


43 

007 




182 


21 


55 


4995 


Tab. 13. Bis. 


182 






The column N shows that 










July, 





21 


55 
21 
42 
17 
10 


the moon is very near her 


I. 




4 


node. There will be a total 


II. 

III. 

IV. 






eclipse of the moon — invisi- 






ble in the United States. 


July, 


1 


3 


25 


Mean 


time ai 


i Gr 


eenw 


ich. 







ECLIPSES 






Full 


® Greenwich time, 


- 


3h. 


25 m 


Albany, Longitude, 


- 


4 


55 


Full 


H§> Albany time, - 


- 


10 


30 



257 

p - M - Chap. II. 



A. M. 

Thus we can compute the time of new or full moon for any 
month in any year ; but, as the numbers for the arguments 
correspond to mean or average motions, and cannot, without 
immense care and labor, be corrected for the true, variable 
motions, the results are but approximate, as before observed. 

ECLIPSES. 

Eclipses take place at new and full moons ; an eclipse of when eciip- 
the sun at new moon, and an eclipse of the moon at full j^ ce 
moon; but eclipses do not happen at every new and full 
moon ; and the reason of this must be most clearly compre- 
hended by the student before it will be of any avail for him to 
prosecute the further investigation of eclipses. 

If the moon's orbit coincided with the ecliptic, that is, if wh - v echp " 

, . , ,. ' ses do not 

the moon s motion was along the ecliptic, there would be an take p ] ace 
eclipse of the sun at every new moon, and an eclipse of the every month 
moon at every full moon ; but the moon's path along the ce- 
lestial arch does not coincide with the sun's path, the 
ecliptic ; but is inclined to it by an angle whose average value 
is 5° 8', crossing the ecliptic at two opposite points on the 
apparent celestial sphere, which are called the moon's nodes. 

If the moon's path were less inclined to the ecliptic, there What would 
would be more eclipses in any given number of vears than ^ e essential 

*- v o ^ ^ for more and 

now take place. If the moon's path were more inclined to whatforfew- 
the ecliptic than it now is, there would be fewer eclipses. er ech P ses - 

The time of the year in which eclipses happen, depends on 
the position of the moon's nodes on the ecliptic ; and if that 
position were always the same, the eclipses would always 
happen in the same months of the year. For instance, if the 
longitude of one node was 30°, the other would be in longi- why an 
tude 30+180, or 210°; and, as the sun is at the first of ecli P se 

. . should take 

these points about the 20th of April, and at the second about p i ace in any 
the 20th of October, the moon could not pass the sun in P articu] ar 
these months without coming very nearly in range with it, of 
course, producing eclipses in April and October. 

17 v* 



258 

Chap. II. 



ASTRONOMY. 



Fig. 52. 



The figure 
represents 
the particu- 
lar paths "of 
the sun and 
moon through 
the heavens. 




For a clearer illustration, we 
present Fig. 52; the right line 
through the center of the figure, 
represents the equator,the curved 
line, T 25 =£= , crossing the equa- 
tor, at two opposite points, re- 
presents the ecliptic, and the 
curved line, Q €> Q, represents 
the path of the moon crossing 
the ecliptic at the points £3 and 
Q; the first of these points is 
the descending, the other, the as- 
cending node. 

As here represented, the as- 
cending node is in longitude 
about 210°, and the descending 
node in about 30°; which was 
about the situation of the nodes 
in the year 1846, and, of course, 
the eclipses of that year must 
have been, and really were, in 
April and October. 

The sun and moon at con- 
junction are represented in the 
figure a little after the sun 
has passed the northern tropic, 
which must be about the first of 
August; and it is perfectly evi- 
dent that no eclipse can then 
take place, the moon running 
past the sun, at a distance of 
about five degrees south; and at 
the opposite longitude the moon 
must pass about five degrees 
north. 

The moon's nodes move back- 
ward at the mean rate of 19° 
19', per year; but the sun moves 



ECLIPSES. 259 

over 19° in about twenty days ; therefore, the eclipses, on Chap - n - 
an average, must take place about twenty days earlier each 
year, or at intervals of about 346 days. 

In May, 1846, the moon's ascending node was in longi- 
tude 216°; in eight years, at the rate of 19° 19', per year, 
it would bring the same node to longitude 61° 28'. The sun 
attains this longitude each year, on the 23d of May, there- 
fore, the eclipses for 1854 must happen in May, and in the 
opposite month, November. 

In computing the time of new and fuU moons, as illustrated The mean ' 
by the preceding examples, the columns marked N, not hith- i umns Nj in 
erto used, indicate the distance of the sun and moon from lhe tables 
the moon's node, at the time of conjunction or opposition. 

The circle is conceived to be divided into 1000 parts, com- Eclipses are 
mencing at the ascending node ; the other node then must hmited t0 a 

° . . , certain space 

be at 500; and when the moon changes within 37 of 0, or along the 
500, that is, 37 of either node, there must be an eclipse of ecliptic - 
the sun, seen from some portion of the earth. When the 
distance to the node is greater than 37, and less than 53, 
there may be an eclipse, but it is doubtful : we shall explain 
how to remove the doubt in the next chapter. 

When the moon fulls within 25 divisions of either 
node, there must be an eclipse of the moon : when the dis- 
tance is greater than 25, and less than 35, the case is 
doubtful ; but, like the limits to the new moon, the 

7 t ' t • Comparative 

doubts are easily removed. We repeat, the ecliptic limits number of 
for eclipses of the sun are 53 and 37 ; for eclipses of the moon, eclipses of 
the limits are 35 and 25. Hence, in any long period of time, moon 
the number of eclipses of the sun is, to the number of eclipses 
of the moon, as 53 to 35. 

In the same period of time, say in one hundred years, there 
will be more visible eclipses of the moon than of the sun ; for 
every eclipse of the moon is visible over half the world at 
once, while an eclipse of the sun is visible only over a very 
small portion of the earth ; therefore, as seen from any one 
place, there are more eclipses of the moon than of the sun. 

In the preceding examples the columns, N, are far within 
the limits, and, of course, there must be an eclipse of the 



260 ASTRONOMY. 

Chap - n - sun on the 26th of May, 1854, and an eclipse of the moon in 
July, 1852. 
How we a s N is in value 9, at the time of new moon, in May, 1854, 

know that an . n - mi i i ». 

eclipse ofthe it shows that the moon will then have passed the ascending 
sun will hap. node, and be north of the ecliptic, and the eclipse must be 
26th oTivia/ visible on the northern portions of the earth, and not on the 

1854, and southern. 

When the moon changes in south latitude, which will be 

circumstance ® ' 

we learn that shown by N being a little more than 500, or a little less than 
u will be an JQ00, the corresponding eclipse, if of the sun, will be visible 

eclipse to . 

some north- on some southern portion of the earth, and not visible in the 

em portion of northern portion; and if of the moon, the moon will run 

through the southern portion of the earth's shadow. 

Table B,p.31, shows the moon's latitude, approximately cor- 

What indi- responding to the column N ; or N is the argument for the 

cates that a latitude, and the heading of the argument columns will 

solar eclipse -iii • t 

will be visi- show whether the moon is ascending to the northward, or de- 
We on some scending to the southward. 

u°on h of *Z The tables from XVI t0 XVnI > together with the solar 
earth. tables, will give approximate values of the elements necessary 

for the calculation of eclipses ; and if accurate results are not 
expected, these tables are sufficient to present general princi- 
ples, and give primary exercises to the student in calculating 
eclipses ; but he who aspires to be an astronomer, must con- 
tinue the subject, and compute from the lunar tables, far- 
ther on. 

The times, and the intervals of time, between eclipses, de- 
pend on the relative motion of the sun and moon, and the 
motion of the moon's nodes. The relative motion of the sun 
and moon is such as to bring the two bodies in conjunction, 
or in opposition, at the average interval of 29 d. 12 h. 44 m. 
3 s., and the retrograde motion of the node is such as to bring 
the sun to the same node at intervals of 346 d. 14 h. 52 m. 
16 s. Neglecting the seconds, and conceiving the sun, moon, 
and node to be together at any point of time, and after an un- 
known interval of time, which we represent by P, sup- 

p 
pose them together again. T nen ~oo7"To~44' represents the 



ECLIPSES. 261 

number of returns of the lunation to the node, m the time Chap. ii. 
P, and the expression . „ , represents the number of of the sun 

and moon in 

returns of the sun to the node in the same time. Each re- relation to 
turn of either body to the node is unity ; therefore, these ex- moon ' s noJe 

, ., , 7 7 7 investigated. 

pressions are to each other as two whole numbers ; say as m 

to n; that is, gg^jj ■ 846TT62 : : m : n > 

_ n m 

(29 12 44)~(346 14 52)' 
Or, - (346 14 52>=(29 12 44> - - - (a) 

n_ 29 1 2 44 

' " m~~346 14 52' 

Reducing to minutes, and dividing numerator and denomi- 

* a ■, n 10631 »•',*■; o ... 
nator by 4, we have — = „^, wo ^ - As this last traction is lr- 
J ' m 124783 

reducible, and as m and n must be whole numbers to answer 
the assumed condition, therefore, the smallest whole number 
for m is 124783, and for n is 10631; that is, as we see by 
equation ( a ), the sun, moon, and node will not be exactly to- 
gether a second time, until a lapse of 124783 lunations, or 
10631 returns of the sun to the same node ; which require a 
period of no less than 10088 years and about 197 days. We 
say about, because we neglected seconds in the computation, 
and because the mean motions will change, in some slight de- 
gree, through a period of so long a duration. 

This period, however, contemplates an exact return to the This period 

... n ,i 7 , 7 ,i t contemplates 

same positions ot the sun, moon, and earth, so that a line ractical im 
drawn from the center of the sun to the center of the moon, possibilities. 
would strike the earth's axis in exactly the same point ; but 
to produce an eclipse, it is not necessary that an exact return . 

to former position should be attained; a greater or less cidencesnev. 
approximation to former circumstances will produce a greater er ha PP en - 
or less approximation to a former eclipse ; but exact coinci- 
dences, in all particulars, can never take place, however long 
the period. 

To determine the time when a return of eclipses may hap- 



262 ASTRONOMY. 

Ch af - tl pen ( particularly if we reckon from the most favorable posi- 

How to tions — that is, commence with the supposition that the sun, 

find the sue- m00Ilj an ^ no( j e are together ), it is sufficient to find the first 

cessive re- ° 

turn of . . 10631 

eclipses. approximate values of the fraction * s}a^q6 ' ^ we ^ n ^ tne 

successive approximate fractions, by the rule of continued 
fractions,* we shall have the successive periods of eclipses, 
which happen about the same node of the moon. 
The approximating fractions are 

1 1 8 L i?_ iM 
H 12 35 47 223" 1831 

a series of These fractions show that 11 lunations from the time an 
siowins the ec ^P se occurs, we may look for another; but if not at 11, it 
periods at will be at 12, and it may be at both 11 and 12 lunations; 
which an( j a £ £ ye Qr g j x i una ^ ong we g^aH fi n( j eclipses at the other 

eclipses oc- x 

cur. node, and the same succession of periods occurs at both 

nodes. 

To be more certain of the time when an eclipse will occur, 
we must take 35 lunations from a preceding eclipse, which 
period is 1033 days 13 h. 40 m., and the sun, at that time is 
about 6° 40' farther from, or nearer to, the node, than before 
— and, if the count is from the ascending node, the moon's 
latitude is about 32' farther south than before, and if from 
the descending node, the moon is about the same distance 
farther north. 

The double of 11, 12, and 35 lunations, from any eclipse, 
may also bring an eclipse. 

If an eclipse occurs within 10° of either node, it is certain 

that eclipses will again happen after the lapse of 47 lunations. 

a brief ex- The period of 47 lunations includes 1387 d. 22 h. 31m., 

aminationof an ^ 4 revolutions of the sun to the node include 1386 d. 

cai return of lib. 29m.; the difference is 1 day 11 h. 29m.; but in this 

eclipses. time the sun will move, in respect to the node, 1° 32 and 
some seconds ; therefore, if the first eclipse were exactly at the 
node, the one which follows, at the expiration of 47 lunations, 

►See Robinson's Arithmetic. 



ECLIPSES. 263 

or 3 years and nearly 11 months afterward, would take place Chap, ii 
1° 32' short of the same node ; and if it were the ascending The Chal _ 
node, the moon's latitude would be about 5' 40" south, and, daean astron- 
if the descending node, about 5' 40" more to the north. ° L mers ca . e , 

° this period 

The period, however, which is most known, and the most Saros. 
remarkable, appears in the next term of the series, which 
shows that 223 lunations have a very close approximate value 
to 19 revolutions of the sun to the node. 

The period of 223 lunations includes 6585.32 days, and 19 
returns of the sun to the same node require 6585.78 days, 
showing a difference of only a fraction of a day ; and if the 
sun and moon were at the node, in the first place, they would 
be only about 20' from the node, at the expiration of this 
period, and the difference in the moon's latitude would be 
less than 2', and therefore the eclipse, at the close of this 
period, must be nearly the same in magnitude as the eclipse 
at the beginning; and hence the expression "a return of the 
eclipse" as though the same eclipse could occur twice. 

This period was discovered by the Chaldsean astronomers, By this pe - 
and enabled them to give general and indefinite predictions I , we 

o o r make a sum- 

of the eclipses that were to happen; and by it any learner, mary predic- 
however crude his mathematical knowledge, can designate the tlon of 

. . eclipses. 

day on which an eclipse will occur from simply knowing the 
date of some former eclipse. The period of 6585 days is 18 
years, including 4 leap years, and 11 days over; therefore 
from any eclipse, if we add 18 years and 11 days, we shall 
come within one day of the time of an eclipse, and it will be 
an eclipse of about the same magnitude as the one we reckon 
from. 

For the purpose of illustrating the method of computing a summary 
lunar eclipses, we wish to find the time when some future inor ™ e 
eclipse of the moon will take place; and from the American time when 
Almanac of 1833, we find that an eclipse of the moon took an t echpse 

' Jl must occur. 

place on the 1st day of July of that year, therefore "are- 
turn of this eclipse " must take place on the 12th of July 
1851. 

By a simple glance into the American Almanac for the 

year 1834, we find a total eclipse of the moon on the 21st of 

18 



264 ASTRONOMY. 

Chap. ji. June — therefore, on the first of July 1852, or at the time 
that the moon fulls, on or about the first of July, there must 
be a large eclipse of the moon, visible to all places from where 
the moon will then be above the horizon ; and furthermore, 18 
years and 11 days after this, that is, in the year 1870, on the 
12th day of July, the moon will be again eclipsed; and, in 
this way, we might go on for several hundred years, but in time 
the small variations, which occur at each period, will gradu- 
ally wear the eclipse away, and another eclipse will as gradu- 
ally come on and take its place. 

In the same manner we may look at the calendar, for any 
year, take any eclipse, that is anywhere near either node, and 
run it on, forward or backward. 

Let us now return to the eclipse of July 12th, 1851. 
Elements To decide all the particulars concerning a lunar eclipse we 
for the com- mugt ]j aYe ^he following data, commonly called elements of 

putation of ° J 

lunar the eclipse : 

eclipses. i The time Q £ ftQj moon 

2. The semidiameter of the earth's shadow. 

8. The angle of the moon's visible path with the ecliptic. 

4. Moon's latitude. 

5. Moon's hourly motion. 

6. Moon's semidiameter. 

7. The semidiameter of the moon and earth's shadow. 
General di- To find these elements, the approximate time of full moon 

obtain the ei° ' lB ^ oun ^ from Table XI, and the tables immediately con- 
ements of nected. For the time thus found, compute the longitude of 
echpses. ^ gun £, om ^ a bi e jy ? an( j ^he tables immediately con- 
nected, as illustrated by examples on page 254. 

Compute, also, the latitude, longitude, horizontal parallax 
semidiameter, and hourly motion in latitude and longitude, 
from the lunar tables, commencing with Table XVI, and fol- 
lowing out the computation by a strict inspection of the ex-- 
am pies we have given ( rules, aside from the examples, would 
be of no avail ) ; and, if the longitude of the moon is exactly 
180° in advance of the sun, it is then just the time of full 
moon; if not 180°, it is not full moon ; if more than 180°, it 
is past full moon. 



ECLIPSES. 265 

It will rarefy, if ever, happen that the longitude of the Chap. n. 
moon will be exactly 180° in advance of the longitude of the 
sun; but the difference will always be very small, and, by 
means of the hourly motions of the sun and moon, the time 
of full moon can be determined by the problem of the couriers* 

The moon's latitude must be corrected for its variation, 
corresponding to the variation in time between the approxi- 
mate and true time of full moon. 

To find the semidiameter of the earth's shadow, where the Role to find 

the semidia- 

moon runs through it, we have the following rule : meter of the 

To the moon's horizontal parallax, add the sun's, and, from earth ' s sha - 

the sum, subtract the sun's semidiameter. 

This rule requires demonstration. Let S (Fig. 53) be 

Fig. 53. 




the center of the sun, E the center of the earth, and Pm a 
small portion of the moon's orbit. Draw p P, a tangent to 
both the earth and sun; from p and P, draw P E and pE, 
forming the triangle p E P. 

By inspecting the figure, we perceive that the three Demonstra, 
angles: tion of the 

SEp+pEP+mEP=lSO°r 

Also, the. three angles of the triangle, P Ep, are, together, 
equal to 180° ; 

Therefore, SEp+p E P+m EP=P+p+p EP ; 

Drop the angle, pEP, from both members of the equation, 
and transpose the angle S Ep, we then have 

mEP=P+p—SEp. 
* Robinson's Algebra — problem of the couriers. 



266 ASTRONOMY. 

Chap - n - But the angle, nuEp, is the semidiameter of the earth's 
shadow at the distance of the moon; SEp is the semidiame- 
ter of the sun ; P, that is, the angle, EPp, is the moon's 
horizontal parallax; and^ is the horizontal parallax of the 
sun ; therefore, the equation is the rule just given.* 
what is rp^ an g} e f ^ moon ' s visible path with the ecliptic is al- 

meant by the ° m m l r 

angle of the ways greater than its real path with the ecliptic, and depends, 
moon's visi- m some measure, on the relative motions of the sun and 

b!e path with 

the ecliptic. m00n ' 

To explain why the real and visible paths of the moon are 
different, let A B (Fig. 54 ) be a portion of the ecliptic, and 
Am a portion of the moon's orbit, then the angle, mAJB, 

Fig. 54. 




b 
is the angle of the moon's real path with the ecliptic. Con- 
ceive the sun and moon to depart from the node, A, at the 
same time, the moon to move from A to m in one hour, and 
the sun to move from A to b in the same time ; join b and m, 
and the angle. mbB, is the angle of the moon's visible path 
with the ecliptic, which is greater than the angle, mA£; 
which is the angle of the moon's real path with the ecliptic. 
On this principle we determine the angle in question. 

All the other elements are given directly from the tables. 



* Some writers have directed us to increase this value of the shadow 
by its one-sixtieth part, but we emphatically deny the propriety of the 
direction. 



ECLIPSES. 



267 



CHAPTER III. 



PREPARATION FOR THE COMPUTATION OF ECLIPSES. 

We snail now go through the computation in full, that it 
may serve for an example to guide the student in computing 
other eclipses. 





Mean N. Moon. 


I. 


ii. 


III. 


IV. 


N. 

431 

511 

43 


1851, 
Six Luna. 
Half Luna. 


• Id. 14h. 21m. 
177 4 24 

14 18 22 


0038 

4851 

404 


3916 
4303 
5356 


40 

92 
58 


39 
95 
50 


. 


193 13 7 
181 


5293 


3575 


90 


84 


985 


As N is within 25 of 1000, 
or 0, there must be an eclipse. 
The sun is 15 short of the as- 
cending node, and the moon at 
full, being opposite, must be 15 
short of the descending node, 
and therefore, in north latitude, 


July, 

I. 
II. 
III. 
IY. 


12 13 7 

3 35 

2 7 

14 

11 


Full © 


12 19 14 


descend 


mg. 






. . .... j 



Chap. m. 

Computa- 
tion of a lu- 
nar eclipse. 

The approx- 
imate time of 
full moon 
computed. 



We now compute the sun's longitude, hourly motion, and 
semidiameter for 1851, July 12, 19 h. 15 m. mean Greenwich 
time, as follows : 







O M. Lon. 


Lon. Peri. 


I. 


II. 


III. 


N. 


s. ° ' " 


s. o / " 




1851 




9 8 32 39 


9 8 22 24 


958 


250 


025 


648 




July 


5 28 24 8 


31 


129 


454 


310 


27 




12 d 


10 50 32 


2 


371 


28 


19 


2 




19 h 
15m 


46 49 
37 




7 










677 


485 


732 


151 




3 18 34 45 


9 8 22 57 


Eq. of center 1 39 38 
I. 10 
IL 18 


318 34 45 




6 10 11 48 


= Mean anomaly. 


III. 20 


O 's hourly motion, 2' 23" 
O's semidiameter, 15' 46" 


3 20 15 11 

Eq. of equinox — 16 


O Ion. 3 20 14 55 





Sun's lon- 
gitude com- 
puted, corre- 
sponding to 
the approxi- 
mate time of 
full moon. 



268 ASTRONOMY. 

Chap. iit. We now compute the moon's longitude, latitude, semidi- 
Direction anieter, horizontal parallax, and hourly motions for the same, 
for comput- mean Greenwich time, as follows : 

ing the 

moon's true 
longitude. FOR THE LONGITUDE. 

1. Write out the arguments for the first twenty equations, 
and find their separate sums. With these arguments enter 
the proper tables ( as shown by the numbers ), and take out 
the corresponding equations, and find their sum. 

2. Write out the evection, anomaly, variation, longitude, 
supplement to node, and the several arguments for latitude, 
in separate columns, corresponding to the given time, and 
write the sum of the twenty preceding equations in the column of 
evection. 

3. Add up the column of evection first ; its sum will be 
the corrected argument of evection, with which, take out the 
equation of evection ( Table XXIV ), and write it under the 
sum of the first twenty equations ; their sum will be the cor- 
rection to put in the column of anomaly. 

4. Add up the column of anomaly, and the sum will 
be the moon's corrected anomaly, which is the argument for 
the equation of the center. With this argument take out the 
equation of the center from Table XXV, and write it under 
the sum of the preceding equations, and find the sum of all, 
thus far. Write this last sum in the column of variation, 
and then add up the column of variation ; which sum is the 
correct argument of variation, and with it take out the equa- 
tion for variation from Table XXVI. 

5. Add the equation for variation to the sum of all the 
preceding equations, and the sum will be the correction for 
longitude, which, put in the column of longitude, and the 
whole added up, will give the moon's longitude in her orbit, 
reckoned from the mean equinox. 

Equation 6. Add the orbit longitude to the supplement of the node, 
oi e equi- ^ ^ g ^ m j g ^ ar or Umen t f reduction to the ecliptic; it 

nox is some- ° r ' 

times called is also the first argument for polar distance. 

nutation m with the argument of reduction take out the reduction 

longitude. ° 

from Table XXVII, and add it to the longitude. 



ECLIPSES. 269 

With argument 19, which is the same as N in the solar ta- Chap. hi. 
bles, take out the equation of the equinox, and apply it ac- 
cording to its sign; the result will be the moon's true longi- 
tude reckoned on the ecliptic from the true equinox. 

FOR THE LATITUDE. 

Add the same correction ( to its nearest minute ) to column General di- 
ll, as was added to the column of longitude, and add its rections lbr 
value, expressed in the 1000th part of a circle, to all the fol- moon > s i at i. 
lowing columns, except column X. Add up these columns, tude - 
rejecting thousands ( or full circles ), and the sums will be 
the 5th, 6th, 7th, 8th, 9th, and 10th arguments of latitude. 

The sum of the moon's orbit longitude, and supplement to 
node, is the first argument of latitude. The sum of column 
II is the second argument of latitude ; the moon's true longi- 
tude is the third argument, and the twentieth of longitude is 
the fourth argument. Then follow 5, 6, &c, up to 10. 
With these arguments enter the proper Tables, and take out 
the corresponding equations, and their sum will be the moon's 
true distance from the north pole of the ecliptic, and, of course, 
will be in north latitude, if the sun is less than 90°, otherwise 
in south latitude. 

N. B. When the first argument of latitude is nearer 6 signs 
than 12 signs, the moon is tending south ; when nearer 12 signs, 
or sign, than 6 signs, it is tending north. 

For the equatorial horizontal parallax. — The arguments for Equatorial 
Evection, Anomaly, and Variation are also arguments for P aralIax and 

° semidiame- 

horizontal parallax, and with these arguments take out the ter depend 
corresponding equations from the tables adapted to this u P on each 

other. 

purpose. 

For the semidiameter. — The equatorial parallax is the ar- 
gument for semidiameter, Table XXXIV. 

For the hourly motion in longitude. — Arguments 2, 3, 4, and General di- 
5 of longitude sensibly affect the moon's motion; they are, L ec ' ons 

o J ' J ' finding tne 

therefore, arguments for hourly motion, Table 36, ( the units hourly mo- 
and tens in the arguments are rejected ). Take out these tl0n of lha 

° . moon. 

equations from table, also take out the equation correspond- 
ing to the argument of evectfon, Table XXXVII. With the 

w* 



270 ASTRONOMY. 

Chap. hi. sum of the preceding equations, at the top, and the corrected 
anomaly at the side, take out the equations from Table 
XXXVIII. Also, with the correct anomaly, take out the 
equation from Table XXXIX. With the sum of all the pre- 
ceding equations at top, and the argument of variation at the 
side, take out the equation from Table XL. Also with 
the variation, take the equation from Table XLI. With the 
argument of reduction take out the equation from Table 
XLIL These equations, all added together, will give the 
true hourly motion in longitude, 
in this pro- For the hourly motion in latitude. — With the 1st and 2d 
portion the arguments of latitude, take out the corresponding quantities 
*L nil™™!? fro m Tables XLIII, and XLIV, and find their algebraic sum, 

the mean mo- ' o ' 

tion of the noting the sign; call the result I. 
moon. Then make the following proportion : 

LI 



32' 56" : L : : I 



32' 56"' 

the true hourly motion in latitude, tending north, if the sign 
is plus, and south, if minus. In this proportion L is the true 
motion of the moon in longitude, and the first term is the 
moon's mean motion ; and the proportion is founded on the 
principle that the true motion in latitude must vary by the 
same ratio as the motion in longitude. 

N. B. In computing the moon's latitude we caution the 
pupil against omitting to add to the arguments II, V, VI, 
VII, VIII, and IX, the same correction as to the column of 
longitude ; its value must be changed into the decimal division 
of the circle for all the columns except column II. 

In the following example the correction for longitude is 
added to column II, and its value to all the following columns 
except column X. 

We find the value in question thus : 

360° : 13° 46' : : 1000 : x. 

The proportion resolved gives x = the number added to 
the several columns. 

But to avoid the formality of resolving a proportion for 
every example, we give the following skeleton of a table that 



ECLIPSES. 271 

may be filled out to any extent to suit the convenience and Chap, m 
taste of the operator. 

Degrees = decimal parts Degrees = parts. 



1 5 


= 


.003 




5 


24 


= 


.015 


1 26 


= 


.004 




7 


12 


= 


.020 


1 48 


= 


.205 




9 





== 


.025 


2 10 


= 


.006 




10 


48 


^ 


.030 


2 31 


— 


.007 




12 


36 


— 


.035 


2 53 


— 


.008 




14 


24 


— 


.040 


3 14 


— 


.009 




16 


12 


= 


.045 


3 36 


= 


.010 












To make 


use 


of this table, we 


will suppose that the cor- 


rection for longitude, in a 


particular example 


is, 11' 


D 31' 25"; 


what is the 


corresponding 


decimal 


or numeral 


part? 


i 


Thus 




9° 


j—^ - 


.030 












2 31 = 


7 









11 31 = .037 

We now continue the examples, hoping to follow these 
precepts. 



272 



ASTRONOMY. 



Chap. III. 



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July, 

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ECLIPSES. 



273 













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274 ASTRONOMY. 

Chap. III. 



The cor- 



ditive. 



The moon's longitude, as just computed, will be 9 20 15 9 
The sun's longitude, at the same time, will be 3 20 14 55 

The difference will be 6 14. 

Therefore, at the time for which these longitudes were 

computed, the moon will be past her full by 14" of arc : to 

correct the time, then, we must find how much time will be 

required for the moon to gain 14" ; which, by the problem 

of the couriers, is 

14 14" 14^ 

- (30.54) — (2.23) . 28' 31" ~ 1711' 

The unit for t is one hour, and the denominator of the frac- 

rection is # m _ 

subtractive tion is the difference of the hourly motions of the sun and 
because the moon, as determined by the tables ; the result is 29 seconds 

moon is past of t j me ^ fee snbtracte(i . 

conjunction, 

otherwise it The Greenwich time will be, 1851, July 12d. 19h. 15m. 0s. 
would be ad- Subtract - 29 

True time of full moon - 12 19 14 31 

But the time given by the lunation table was 19 h. 14 m., 
differing only 31 seconds from the true time ; the approxi- 
mate and true time, however, do not commonly coincide as 
near as this ; if they did, none but the most rigid astrono- 
mer would use the lunar tables for the time of conjunction or 
opposition. 

To be very exact we must correct the moon's latitude for 
what it will vary in 31 seconds ; that is, in this case, increase 
it 4".5. The moon's latitude, at the time of full moon, is, 
therefore, 42' 53".4. 

We have now all the elements necessary for computing the 
eclipse, or, at least, we have all the materials for finding 
them, and, for convenience, we collect the elements together : 

d. h. m. s. 

1. True time of full moon, July, - - 12 19 14 31 

2. Semidiameter of earth's shadow 
(page 265), - - - ■ ° 39' 39" 

3. Angle of the moon's visible path 
with the ecliptic, * - - 5 38 26 



* This is the angle of the base of a right-angled triangle, whose base 



ECLIPSES. 



275 



4. Moon's latitude N. descending, - - 42 53.4 

5. Moon's hourly motion from the sun, - 28 31 

6. Moon's semidiameter, - - - 15 4 

7. Semidiameter of £> and earth's shadow, 54 43 
Whenever the moon's latitude, at the time of full moon, is 

less than this last element, the moon must be more or less 
eclipsed ; and it is by computing and comparing these two ele- 
ments, viz., 4 and 7, that all doubtful eases are decided. 

TO CONSTRUCT A LUNAR ECLIPSE. 

From any convenient scale of equal parts, take the 7th ele- 
ment in your dividers (54 43) = 54f , and from C, as a center 
with that distance, describe the semicirele B D HE (Fig. 55). 
Take A = the 2d element, and describe the semidiameter 
of the earth's shadow. From C, the center of the shadow, 
draw On at right angles to B E, the ecliptic, above BE, when 
the latitude is north, as in the present example, but below, 
if south. 

Fig. 55. 



Chap. in. 



When the 
moon has 
very little la- 
titude de- 
scribe a full 
circle. 

When large 
south lati- 
tude, de- 
scribe only 
the lower 
semicircle. 




Take the moon's latitude from the scale of equal parts, 
and set it off from C to n. Through n draw DnB, the 
moon's path, so that the line shall incline to BE, the ecliptic, 
by an angle equal to the 3d element. Conceive the moon's 

is the hourly motion of the moon from the sun (28' 31"), and the per- 
pendicular, the moon's hourly motion in latitude ( 2' 49" ). See 
page 266, figure 54. 



276 ASTRONOMY. 

Chap, m center to run along the line from D to H, and from C draw 
Cm perpendicular to DH. 

When the moon is ascending in her orbit, DH must incline 
the other way, and Cm must lie on the other side of Cn. 

The eclipse commences when the moon arrives at D. It is 
the time of full moon when it arrives at n ; the greatest ob- 
scuration occurs when it arrives at m, and the eclipse ends at 
H The duration is the time employed in passing from D to 
H: and to find the duration apply DH to the scale, and thus 
The 5th eie- find its measure. Divide this measure by the 5th element, 
i s ie an( j we gjjgjj h ave the hours and decimal parts of an hour in 

moons angu- ->- 

lar motion the duration. Also apply Dn to the scale and find its mea- 
from the sun. sure Divide this measure by the 5th element, for the time 
of describing Dn, also divide the measure nlf {or the time of 
describing nH. 

The time of describing Dn, subtracted from the time of 
full moon, will give the time of the beginning of the eclipse, 
and the time of describing nH, added to the time of full 
moon, will give the time when the eclipse ends. 

With lunar eclipses the time of greatest obscuration is the 
instant of the middle of the eclipse, provided the moon's mo- 
tion from the sun, for this short period of time, is taken as 
uniform, as it may be without sensible error. 

In reference to this example Dn = 3V and nH=39'. 
These distances, divided by 28' 31", give 1 h. 5 m. 16 s. for the 
time of describing Dn, and lh. 22m. 4s. for nH: whole 
time, or duration, 2 h. 27 m. 20 s. 

h. m. s. 

Astronomi- 
cal time con- Therefore from the time of full $ 19 14 31 
verted into Subtract - - - 1 5 16 

civil time. 

Eclipse begins - - - 18 9 15 

Add the duration - - 2 27 20 

Eclipse ends - - - 20 36 35 

This eclipse 

Earope aud That is, in 1851, July 12 d. 18 h. 9 m. 15 s., mean astrono- 

why. mical time, the eclipse begins ; but this time corresponds with 

July 13, at 6 h. 9 m. in the morning, and at this time, the sun 

will be above the horizon of Greenwich, and, of course, the 



ECLIPSES. 



277 



full moon, which is always opposite to the sun, will be below Chap. m. 
the horizon, and the eclipse will be invisible to all Europe. Visible . 

In the United States, however, the eclipse will be visible, theu.s. 
for, at these points of absolute time, the sun will not have 
risen nor the moon have gone down ; but, to be more definite, 
we demand the times of the beginning, middle, and end of the 
eclipse, as seen from Albany, N. Y. To answer this demand, 
all we have to do, is to subtract from the Greenwich time the 
difference of meridians between the two places, which, in this 
case, is 4h. 55 m. ; and the result is, 

Beginning of the eclipse 13 d. 1 h. 14 m. morning, 

Middle ---- 2 28 

End of the eclipse - - 3 41 „ 

In the same manner we would compute the time for any 

other place. 

For the quantity of the eclipse we take the portion of The quan- 

the moon's diameter, which is immersed in the shadow, tlty of the 

eclipse how 

at the time of greatest obscuration, and compare it with found, 
the whole diameter of the moon; and in the present ex- 
ample, we perceive, that not quite half of the diameter is 
eclipsed — about 5 digits when the whole is called 12, or 0.4 
when the diameter is 1. 

All these results, however, except the time of full moon, 
are approximate, because we cannot, nor do we pretend to 
construct to accuracy ; but any mathematician can obtain accurate 
results by means of the triangles D C H and Cnm, and the 
relative motion of the moon from the sun. 

In the right-angled triangle Cnm, right-angled at m, On The exact 
is the latitude of the moon = 42' 53".4 = 2573".4, and the computation 
angle n Cm = 5° 38' 26" ; with these data we find mn= tion of th a e 
253", and Cm = 2561".6. eclipse. 

In the right-angled triangle C Dm, or its equal CmE,we 
have - - Cm 2 -\-mH 2 = CIT 2 ; 

Or, - - mH 2 = CH 2 — Cm 2 ; 

Or, - - mH 2 =(CIT-\-Cm) (CH—Cm). 

CITis the 7th element = 3283", and Cm == 2561".6. 

Therefore, m ff= V (5844.6 ) (721.4) = 2043".4. This 

x 



278 ASTRONOMY. 

Chaf - m - divided by 1711", the 5th element, gives the time of half 
the duration of the eclipse 1 h. 12 m. ; therefore the whole du- 
ration is 2h. 24 m., which is 3 m. 20 s. less than the time we 
obtained by the rough construction. 

The distance nm, as just determined, is 253", and the time 
of describing this space, at the rate of 1711" per hour, re- 
quires 8 m. 52 s., which taken from, and added to the semi- 
duration, gives 1 h. 3 m. 8 s. from the beginning of the eclipse 
to full moon, and 1 h. 20 m. 52 s. from the full moon to the 
end of the eclipse. 
The trigo- ]? or t ne magnitude of the eclipse we add the moon's semi- 
compntation diameter in seconds ( 904" ) to Cm ( 2561".6 ), and from the 
ofthemagni- sum subtract the semidiameter of the shadow in seconds 
eclipse * ( 2379 ), and the remainder is the portion of the moon's di- 
ameter not eclipsed. Subtract this quantity from the moon's 
diameter and we shall have the part eclipsed. Divide this 
by the whole diameter and the quotient is the magnitude of 
the eclipse, the moon's diameter being unity. 

Following these directions we find the magnitude of this 
eclipse must be 0.397. 
The con- j n a jj $j es@ computations we were guided bv the construc- 

struction a L ° # * 

sufficient tion ; which will always prove a sufficient index, and all that 
guide to car- should be required. 

trigonometru ^ e ma y determine, m an y case > whether the eclipse will or 
cai computa- will not be total, by the following operation : 

Subtract the €>'s semidiameter from the semidiameter of 
the shadow, and if the moon's latitude, at the time of full 
moon, is less than the remainder, the eclipse will be total, 
otherwise not. 

To find the duration of total darkness. — Diminish the semi- 
diameter of the shadow by the semidiameter of the moon, and 
from the center of the shadow describe a circle, with a radius 
equal to the remainder; a portion of the moon's path must 
come within this circle ; that portion, measured or divided by 
the hourly motion, will give the time of total darkness. 

When the moon's latitude is north, as in the present ex- 
ample, the southern limb of the moon is eclipsed — and con- 
versely. 



ECLIPSES. 279 



CHAPTER IV. 

S0LAK ECLTPSES GENERAL AND LOCAL. 

The elements for a solar eclipse are computed in the same Chap. iv. 
manner as the elements of a lunar eclipse ; all of which are General ai- 
found by the solar and lunar tables. rections to 

_, . .-■■ find the ele- 

J he approximate time of new moon is first computed, and ments . 
for this time, compute the sun's longitude, declination, paral- 
lax, semidiameter, and hourly motion; and for the same 
time compute the moon's longitude, latitude, hourly motion in 
longitude and latitude, horizontal parallax, and semidiameter. 

If the longitudes of both sun and moon are found to be the 
same, then the approximate time of conjunction ; found by the 
lunation tables, is the same as the true time ; if not, we pro- 
portion to the true time, as described in the last chapter. 

The elements for a general solar eclipse are : 

I. The time of £ * at some known meridian. 2. Longi- what eie- 
tude of O and f). 3. Q's declination. 4. f)'s latitude. ments are 

^ w necessary. 

5. Q's hourly motion. 6. C's hourly motion in longitude. 
7. #)'s hourly motion in latitude. 8. The angle of the ®'s 
visible path with the ecliptic. 9. #)'s horixontal parallax. 
10. C>'s semidiameter. 11. o' s semidiameter. 12. ©'s 
horizontal parallax. 

For a local eclipse, the latitude of the particular locality 
must also be given, or considered as one of the elements. 

As we can best illustrate general principles by taking a a definite 



particular example, we now propose to show the general course e **^ p e pr ° 
of an eclipse of the sun, which will occur in May 1854; where 
it will first commence on the earth ; in what latitude and longi- 
tude the sun will he centrally eclipsed at noon, and where ; in 
what latitude and longitude the eclipse will finally leave the earth. 

We speak of an eclipse of the sun being on the earth; by Some gene- 
this we mean the moon's shadow on the earth. If an observer „ pre ^ 

nary expJa- 

is in the moon's shadow, of course, the sun would be in an nations, 
eclipse to him; and, if a tangent line be drawn between the 

* Sign of conjunction. 



280 ASTRONOMY. 

chap. iv. sun and moon, and that line strike the eye of an observer on 
the earth, to that observer the limbs of the sun and moon 
would apparently meet, and all projections of eclipses are on 
the principle of lines drawn from some part of the sun to 
some part of the moon, and those lines striking the earth. 
When no such lines can strike the earth there can be no 
eclipse. For the sake of simplicity in explaining a projection 
Point of °f a s °l ar eclipse, whether it be general or local, an observer 
view. is supposed to be at the moon, looking down on the earth, 

viewing the moon's shadow as it passes over the earth's disc, 
and, of course, the earth to him appears as a plane, equal to 
the moon's horizontal parallax. 

The approximate time of new moon will be found com- 
puted on page 254, and, if very close results are not required, 
we may compute the sun's longitude, declination, hourly mo- 
tion, and semidiameter for this time, and take out the moon's 
horizontal parallax, hourly motion, and semidiameter from 
Table IX ; but we have computed the elements more accu- 
rately by the lunar tables, and find them as follows : 

d. h. m. s. 

1. Greenwich mean time of d 1854, May 26 8 45 39 

Accurate 2. Lon. of Q and C - - - 65° 14' 6" 

elements for 3 D ec]ination of the Q m %1 11 43 N. 

the solar ^ 

eclipse, 4. Latitude of the if) - 

which will 5 > s hourl motion in 1q _ 

take place J 

May 26, 6. #) s hourly motion in Ion., - 

1854, 7, ^)' g hourly motion in lat., tending north, 

From 5, 6, and 7 we obtain 8, as explained 
in the last chapter. 

8. Angle of the moon's visible path 

with the eclip., - 

9. The €>'s horizontal equatorial parallax, 

10. The D's semidiameter, 

11. The O's semidiameter, 

12. The 0' s horizontal parallax, always taken at 
Add together the O's horizontal parallax, the #)'s hori- 
zontal parallax, and the semidiameters of O aQ d O, and if 
the moon's latitude is less than this sum, there will be an 





21 


19 N. 




2 


24 




30 


3 


I 


2 


46 


o 


i 


n 


5 


42 


50 




54 


30 




14 


51 


at 


15 


48 
9 



ECLIPSES. 281 

eclipse, otherwise not; and it is by comparing this sum with Chap. iv. 
the moon's latitude that all doubtful cases are decided. 

TO CONSTRUCT A GENERAL ECLIPSE. 

1. Make, or procure, a convenient scale of equal parts, and 
from any point as C ( Fig. 56 ) with the radius C B, equal to 
the sum of the horizontal parallaxes of O and Q ( in the pre- 
sent example 54' 39", the minute is the unit ), describe the 
semicircle C B P H> or the whole circle, when the case re- 
quires it. When the moon has small latitude (less than 20') 
describe the whole circle ; when the moon has large north lati- 
tude describe the northern semicircle, when south describe the 
southern semicircle. 

Through C draw VCD PL perpendicular to HB. This 
perpendicular will represent the plane of the earth's axis, as 
seen from the moon. 

From P take PA, P F, each equal to the obliquity of the 
ecliptic 23° 27' 30", and draw the chord A F. 

On A F, as a diameter, describe the semicircle ALF. , A ., 

find the axis 

2. Find the distance of the sun from the tropic, nearest to of the eciip- 
it, by taking the difference between the sun's longitude and tlc ' 

90° or 270°, as the case may be. In the present example we 
subtract 65° 14' from 90°, the remainder is 24° 46'. Take 
L T, equal to 24° 46', and draw TE parallel to L C. Draw 
C E the axis of the ecliptic. 

By the revolution of the earth round the sun, the axis of The axis 
the ecliptic appears to coincide with the axis of the equator, ° ic ^^m 
when the sun is at either tropic, and it appears to depart in position, 
from that line by the whole amount of the obliquity of the 
ecliptic ; and the time of this greatest departure is when the 
sun is on the equator. That is, CE runs out to C A at the 
vernal equinox, and runs out to C F at the autumnal equi- 
nox. As a general rule, CE, the axis of the ecliptic, is to 
the left of CP, the axis of the equator, from the 20th of De- 
cember to the 20th of June, and to the right of that line the 
rest of the year. Draw C O the axis of the moon's orbit, so How t0 fin d 
that the angle O CE shall be equal to the angle of the th & e £"* 0I r 
moon's visible path with the ecliptic, and C O is to the left of Mt. 

x* 



tor. 



-S2 ASTRONOMY. 

chap. iv. CU when the eclipse is about the ascending node, as in this 
example, but at the right when the eclipse is about the de- 
cending node. 

For this projection to appear natural, the reader should 
face the north, so that H will appear to the west, and B on 
the east of the figure. 

The shadow of the moon across the earth is from a western 
to an eastern direction, therefore, the moon is conceived to 
come in on the earth from the west side. 
The equa- rp^e point, C, is perpendicular to the sun's declination, and 
C Vis the sine of the declination, and the curved line, HVB, 
is a representation of the equator, as seen from the moon. 
When the sun has no declination, the equator draws up into 
a straight line. 
How to 3. Take C n from the scale of equal parts, making it equal 
draw the j. Q ^ moon ' s latitude, and through the point n, and at right 

naoon's path. - . . 

angles to C O, draw the line klmnrpe, which represents the 
center of the shadow, or the moon's path across the disc. 

From C, as a center, at the distance C 0, describe the 
outer semicircle, equal to the sum of the moon's horizontal 
parallax, the sun's horizontal parallax, and the semidiameter 
of both sun and moon ; then If is the semidiameter of the 
sun and moon. 

When the eclipse first commences, the center of the moon 
is at k, and the center of the sun is on the circumference of 
the other circle, in a direct line to C, not represented in the 
figure, therefore, the two limbs must then just touch. 

As C is the center of the earth, and H on the equator, 
therefore CH is a line in the plane of the equator, and the 
point, k, is a little below the equator ; which shows that the 
eclipse first commences on the earth a little south of the 
equator. 
How to de- The time that the eclipse is on the earth is measured by 

dn™tion of a * ne ^ me required for the moon to pass from k to e with its 

general true angular motion from the sun. 

echpse. rpj^ j^g^ f this line, k e, can be found from the ele- 

ments, and trigonometry, as in an eclipse of the moon, and 
the time of describing it is found in the same way. 



ECLIPSES 



283 




Fig. 56, 



284 ASTRONOMY. 

chap. iv. When the moon's center conies to I, the central eclipse 

Howtode- commences, and the arc, HI, shows that it must be about in 

termine in t k e latitude of 7° north. When the moon's center comes 

what lati- 
tudes the to r, the sun will be centrally eclipsed at apparent noon; and 

eclipse will (j r j s the sine of the number of degrees north of the sun's 

enter, pass -,.. . •,.-,. , . . ■> of> 

over, and decimation, which, in this case, is about 23° ; hence to the 
pass off the sun's declination, 21° 12', add 23°, making 44° 12'; showing, 
as near as a mere projection can show, that the sun will be 
centrally eclipsed at noon on some meridian, in latitude 44° 12' 
north. The central eclipse will end, or pass off the earth, 
when the moon's center arrives at p, and the arc, Bp, from the 
equator, shows that the latitude must be about 41° north. The 
eclipse will entirely leave the earth when the moon's center 
arrives at e, and for its limb to touch the sun, the sun's cen- 
ter must be at h, and the arc, B h, shows that the latitude 
must be about 30° north. 

The lines, cd and ab, parallel to the moon's path, and dis- 
tant from it equal to the sum of the semidiameters of sun and 
moon, represent the lines of simple contacts across the earth, 
or limits of the eclipse ; cd is the southern line of simple con- 
tact, and a b is the northern line of simple contact, and the 
latitudes at which these lines make their transits over the 
earth, are determined precisely as the latitudes on the cen- 
tral line. 
We may J3ut we need not stop at coarse approximations, we have 
rate compn- a ^ tne data for correct mathematical results, on the same 



tations by principles as we determined those in relation to a lunar eclipse. 

plane tr' 
nometry. 



In the triangle, Cnr, we have the side, Cn, the moon's 



latitude in seconds, which may be used as linear measure, as 

yards or feet, and in proportion thereto, we may compute Cr 

and nr, when we knoiv the angle, n Cr. 

An equa- jj u t the following equation always gives the tangent of the 

position of angle, E CD, or n Cr, calling the sun's distance from the sol- 

the axis of gtice j) t the obliquity of the ecliptic E, and the radius, unity. 

the ecliptic. 

tan. EC £=ta,n. E sin. D* 

* The student who has acquired a little skill in analytical trigono- 
metry can discover the preliminary steps to this equation; the princi- 
ples are all visible in the construction of the figure. 



ECLIPSES. 285 

To the angle, E CD, add the angle, G CE, the angle of the Chap. iv. 
moon's visible path with the ecliptic, and we have the whole 
angle, G C D, or m Cr. Cmn is a right angle, and in the 
two triangles, Cmn and Cmr, we have all the data, and can 
compute n r and r C. 

When the moon arrives at m, it is in the line of conjunction 
in her orbit ; when it arrives at w, it is in ecliptic conjunction; 
and when it arrives at r, it attains conjunction in right as- 
cension. 

For the last six or eight years, the English Nautical Al- Recent 

i • ,1 • , • i •,• ••!, changes in 

manac has given the conjunctions and oppositions m right as- the English 
cension, in place of conjunctions and oppositions in longitude, Nautical Al- 
and has given the difference of declinations between the sun 
and moon, in place of giving the moon's latitude ; that is, it 
has given the time that the moon arrives at r, in place of n, 
and given the line, Cr, in place of Cn. 

x^.11 lunar tables give the ecliptic conjunction at n, and from 
this we can compute the time at r, by means of the triangle, 
Cnr. 

Having explained the principle of finding the latitude on 
the earth, when a solar eclipse first commences, we are now 
ready to show another important principle — how to find the 
longitude ; and with the latitude and longitude, we have the 
exact point on the earth. 

Where an eclipse first commences on the earth, it com- The method 
mences with the rising sun, and finally leaves the earth with ]° on J| tl ^f 
the setting sun. In this example, we have decided that the where the 
eclipse must commence very near the equator, not more than ™^** ^ 
one degree south ; but in that latitude the sun rises at 6 h., earth. 
a. m., apparent time ; therefore, at the place where the eclipse 
commences, it is six in the morning, apparent time. 

From the scale of equal parts, take the moon's hourly mo- 
tion from the sun in the dividers (27' 39"), and apply it on 
the linekq, it will extend three times, and a little over, to the 
point n. This shows that three hours, and a little more ( we 
say 3h. 3 m.) must elapse from the first commencement of 
the eclipse to the change of the moon at n. Hence, by the 
local time at the place of the commencement of the eclipse, 



286 ASTRONOMY. 

Chap. iv. the moon changes at 9 li. 3 m. in the morning, apparent time ; 
but the apparent time of new moon at Greenwich is 8 h. 49 m., 
p. m., making a difference of 11 h. 46m., for mere locality; 
the absolute instant is the same; the difference is only in 
meridians which correspond to a difference of longitude of 
175° 30'; and it is west, because it is later in the day at 
Greenwich. 

The method rj^g cen t ra i eclipse also first comes on the earth at a place 

of finding x ^ * 

where the where the sun is rising. In this example it first strikes the 
central earth at the point I, in latitude about 7° N. ; but, in latitude 

eclipse first 

strikes the 7° N., and declination 21° N., the sun rises at 5h. 48 m., 
earth. A# M#j apparent time ( Prob. II ), and from that time to the 

change of the moon, namely, the time required for the moon 
to move from I to n, is ( as near as we can estimate it by the 
construction ), 1 h. 56 m., therefore, the time of new moon, in 
the locality where the central eclipse first commences, is 7 h. 
44 m. in the morning. From this to 8 h. 49 m. in the even- 
ing, the time at Greenwich, gives a difference of 13 h. 5 m., 
reckoned eastward from the locality; orlOh. 55m., reckoned 
westward ; which corresponds to 196° 15' west longitude from 
Greenwich, or 163° 45' east longitude; the meridian is the 
same. If the longitude is called east, the day of the month 
must be one later ; but, to avoid this, we had better call the 
longitude west. 
To find the Where the sun is centrally eclipsed on the meridian, it is 

°h g r " 6 h J us * ^' a PP aren * ti me > * ne moon's center is then at r, and, 
sun will be by the construction, it must be about seven minutes after 
centrally conjunction in that locality ; hence, the conjunction is seven 

eclipsed at J ... 

noon. minutes before 12, and at Greenwich it is 8 h. 49 m. after 12, 

giving 8 h. 56 m. for difference of longitude, or 134° west 
longitude. 

The central eclipse will leave the earth with the setting 
sun, when the center of the moon and sun are both at p ; but 
the latitude of p we decided to be 40° north, and in this 
latitude, when the sun's declination is 21° ir, as it now 
is, the sun sets at 7h. 15m. apparent time; but this is 
lh. 40 m. after conjunction, therefore, the conjunction, in 
that locality, must be at 5 h. 35 m. ; but, at Greenwich, it is 



App. time Gr. 


Lat. 




Longitude, 

o / 


m, 5 46 
6 53 


1 s. 

7 N. 


175 30 W. ResuItsme - 

-tt\rt t r ttt chanically 
196 15 W. tak en from 


8 56 


46 


134 00 W. the P'ojec- 


10 34 


40 


48 30 W. tion - 


1146 


30 


73 30 W. 



ECLIPSES. 287 

8h. 49 m., giving, for difference of longitude, 3h. 14 m., or Chap. iv. 
48° 30' west. " " 

The eclipse finally leaves the earth in latitude 46° north ; To find the 
but, in this latitude, the sun sets at 6 h. 51 m., and the con- lo ^ ltude 

7 where toe 

junction will be 3h. 0m. sooner (the time required for the eclipse will 
moon to pass from n to q ), therefore the conjunction, in this leave the 
locality, must be at 3h. 51m.; but, at Greenwich, it will be 
8h. 49 m., giving 4h. 58 m. for difference of longitude, or 
74° 30' west. 

Thus, by the mere geometrical construction, we have 
roughly determined the following important particulars : 



Eclipse commences, May 26, 
Cen. eclipse commences, 
Cen. eclipse at local noon, 
Cen. eclipse ends, 
End of eclipse, 

To find the latitude of the first commencement of simple The loca!i - 
contact on the southern line, all we have to do is, to find the southern and 
aro, Be, and for the latitude on the northern line, we find the northern 
arc, Ha : the point, c, is in latitude about 27° south, and a in 1 , nea 

' * ' ' ' pie contact. 

about 54° north. 

The southern line of simple contact leaves the earth at d, 
between the seventh and eighth degrees of north latitude, and 
the northern line passes off beyond the pole. 

We have, thus far, taken the results but approximately 
from the projection, and the projection is sufficient to teach 
us principles ; and it must be our guide, if we attempt to ob- 
tain more minute results ; and with the elements and the figure 
we have the whole subject before us as minutely accurate 
as it is magnificent, and as simple as it is sublime. 

To complete our illustration, we now go through the trigo- 
nometrical computation. 

In the triangle, Cnm, we have Cn—2V 19"=1279, the 
angle, m Cw=5° 42' 50", and the angle, m, a right angle. 

Whence, 0771=1273", and mrc=127".3. 



288 ASTRONOMY. 

chap, iv . tan. E CD=n CV=tan. (23° 27' 32") sin. (24° 45' 54") 
in these ( page 284). 

ZIZ Whence, E CD= 10° 18' 8", 

moon-, lati. Add, G G E= 5° 42 ' 50", 

nule and the ' 

distancea Sum is G 2=m Cr=16° 0'58". 

l^c^Te In the trian g le m Cr > we have Cm ( 12 73), the perpendicu- 
circumfer- lar, and the angle m Cr, as just determined ; whence, 

^Tnel 6 ^' mr=365".S ; (7r=1324".3. 

In the triangle, Cmp, Cp is the horizontal parallax of 
moon and sun (54' 30")+9", or, 54' 39"=3278". 

By the well-known property of the right-angled triangle, 

Cm 2 -\-mp 2 = Cp 2 . 
Or, mp 2 = Cp 2 — Cm 2 =(Cp-{-Cm) (Cp—Cm), 



That is, m j p= N /(4551)(2005)=3020".7. 

Therefore, Ip, the whole chord, is 6 041 "A, which, divided 
by 1659" (the moon's motion from the sun), gives 3.646 h., 
or 3 h. 38 m. 46 s., for the time that the central eclipse will 
he on the earth. 

In the same manner the line, m q, is found. 

That is, m q= J(lTq-\-Cm) ( Cq—Cm), 

But, Cg=54' 39"+14' 5.1"+15' 48"=5118". 

Or, m q= 7(6391)(3845)=4957".3. 

Therefore, the whole chord, kq, is 9814.6, which, divided by 
1659", gives 5 h. 58 m. 34 s., for the entire duration of the 
general eclipse on the earth. 

On the supposition that the moon's motion from the sun is 
uniform for the six hours that the eclipse will be on the earth, 
the several parts of the moon's path will be passed over by 
the moon, as follows : 



Accurate From & to Z in 1 h. 9 m. 54 s 

results on the 
condition of 

invariable el- From m to n in 4 36 to rj in ecliptic 

ements. 



From 


I to m in 


1 


49 


23 


From 


m to n in 




4 


36 


From 


n to r in 




8 


37 


From 


r top in 


1 


36 


10 


From 


p to q in 


1 


9 


54 



to £ in orbit, 
to 6 in eclipt 
to d in right ascension. 



8 k 


49 m. 


Os. 


3 


3 


53 


5 


45 


7 


6 


55 


1 


8 


57 


37 


LO 


33 


47 


LI 


43 


41 



ECLIPSES. 289 

The apparent time of ecliptic conjunction, at Greenwich, chap. rv. 
as determined by the tables (and applying the equation of 
time), is at 
Subtract from k to ecliptic ^ , 

Eclipse commences, Greenwich app. time, 
Central eclipse commences (add 1 9 54), 
Sun centrally eclipsed on some meridian, or 
£ in right ascension, Greenwich time, 

at (add 2 2 36), 
Central eclipse ends at (add 1 36 10), 
End of eclipse at (add 1 9 54), 

By comparing these times with those obtained simply by a careful 
the projection, we perceive that the projection is not far out P r °J eclj on 
of the way, notwithstanding the terms rough and roughly that rate than is 
we have been compelled to use concerning it. Indeed, a good generally 
draftsman, with a delicate scale and good dividers, can decide snppos 
the times within two minutes, and the latitudes and longitudes 
within half a degree; but all mathematical minds, of course, 
prefer more accurate results; yet, however great the care, 
absolute accuracy cannot be attained ; the nature of the case 
does not admit of it.* 

To find whether the point h is north or south of the equa- 

*The astronomer, by making use of his judgment, can be very ac- 
curate with very little trouble; he perceives, at a glance, what ele- 
ments vary, and what the effects of such variation will be, but a learner, 
who is supposed not to be able to take a comprehensive view of the 
whole subject, must go through the tedious process of computing the 
elements for the times of the beginning and end of the eclipse, as well 
as the time of conjunction, if he aims at accuracy, but an astronomer 
can be at once brief and accurate. In computing the moon's longi- 
tude, in the present example, the astronomer would notice in particu- 
lar the moon's anomaly, and, by it, he perceives whether the moon's 
hourly motion is on the increase or decrease, and at what rate. 

It is on the decrease, and the first part of the chord k in is passed over 
by the moon in about 7 seconds less time than our computation 
made it, and the last part requires about 7 seconds longer time ; but 
the times of passing m and n should be considered accurate, and the 
times of beginning and end should be modified for the variation of 
the moon's motion, making the beginning and end 7 seconds later, and 
the beginning and end of the central eclipse about 4 seconds later. 
19 Y 



290 ASTRONOMY. 

Chap. iv. tor, we conceive k and C joined, and if the angle m Ck is 
greater than the angle m CH, the point k is south, otherwise 
north. 

By trigonometry, Ck : km : : Bine 90° : sine mOk; 

O I II 

Or, 5118 : 4957".3 : sin. 90 : sin.™ Ck=1b 35 20 
To this add G CD, - - - 16 58 

Sum is the angle r Ck - - - 91 36 18 

This angle shows that the eclipse will first touch the earth 
in latitude 1° 36' 18" south. 

To find the arc HI, conceive the points CI joined, and the 
two triangles Clm, m Cp are equal. 

&nd CI : I'm : : sin. 90° : m CI; 

O I II 

Or, 3278 : 3020.7 : : sin. 90 : sin. m C7=67 7 50 
To this add Q CD, - - - 16 58 

The sum is, - - - 83 8 48 

Where the This angle shows the latitude of the point I to be 6° 51' 
strikes th "' -^" nort, h. That is, the central eclipse first touches the 
earth earth in 6° 51' 12" of north latitude; differing very little from 

the point determined by construction. 

To find the latitude of the point p, we have mCl = m Cp 
= 67° 7' 50", and subtracting 16° 0' 58", we have the 
polar distance, or co-latitude; the result is, that the central 
eclipse passes off at latitude 38° 53' 8" north, and the gene- 
ral eclipse entirely leaves the earth in latitude 30° 25' 38". 

To find the latitude of the point r, we consider Cr to be a 
sine of an arc, and C P the radius. 

Therefore, 3278" : 1324".3 : : R : sin. * = 23 49 50 
To this add the sun's declination, - 21 11 43 

Sum is latitude where the sun will be 

centrally eclipsed on the meridian, - 45 1 33 N. 
How to find Wherever the sun is centrally eclipsed on the meridian, it 
the longitude is apparent noon at that place, but at Greenwich the apparent 
of the place ^ q is g ^ 57 m 37 g ^ p M ^ . ^ g difference, changed into Ion- 

wrier© in 6 

sun is central- gitude, gives 134° 25' west, within a degree of the result de- 
iy eclipsed on termined from the projection; and it is not important to go 

the meridian. , . 

over a tngonometncal computation tor the longitudes, since 



ECLIPSES. 291 

we are sure of knowing how to do it; and we are also sure chap. iv. 
that the results will not differ much from those already de- 
termined. 

In short, from the elements, the figure, and a knowledge sufficient 
of trigonometry, we can determine all the important points in J ala in the 
each of the three lines cd, kg, and a b, for between them we 
have, or may have, a complete net-work of plane triangles. 



CHAPTER V. 

LOCAL ECLIPSES, ETC. 



We now close the subject of eclipses by showing how to Chap. v. 
project and accurately compute every circumstance in rela- 
tion to a local eclipse. 

For an example, we take the eclipse of May, 1854, and for 
the locality, we take Boston, Mass., because we anticipated a 
central eclipse at that place, but the result of computations 
shows that it will not be quite central even there. We use 
the same elements as for the general eclipse. 

THE CONSTRUCTION. 

Draw a line CD, and divide it into 65 equal parts, and The scale, 
consider each part or unit as corresponding to one minute of 
the moon's horizontal parallax. From 0, as a center, at a 
distance equal to the horizontal parallax of the sun and moon 
( 54 39 ), describe a semicircle north or south according to 
the latitude, or describe a whole circle, if the latitude is near 
the equator. 

From C draw (7gs, the universal meridian, at right angles 
to CD, and from 25 take go T and 25=£=, each equal to the 
obliquity of the ecliptic ( 23° 27' ) and draw the straight line 
°p^=, T on the right. Subtract the sun's longitude from 
90° or 270° to find its distance from the nearest solstitial 
point, and note the difference (in this example 24° 46'). fle */ * 

From the point, a, with a T, as radius, make a O, equal to ecliptic. 



292 ASTRONOMY. 

Cg^-T- the sine of 24° 46',* and join C 0, and produce it to E; CE 
is the axis of the ecliptic ; this line is variable, and is on the 
other side of the line, C<g>, between June 20, and Decem- 
ber 21. 
How to find F rom E take the arc, EL, equal to the moon's visible path 
tne moon's w ith the ecliptic, to the right of E, when the moon is descend. 
o-bit. ing, but to the left, when ascending, as in the present exam- 

ple. Join C L, a line representing the axis of the moon's orbit. 
To and from the reduced latitude of the place add and sub- 
tract the sun's declination: 

Thus, Boston, reduced latitude, - 42° 6' 39" N. 

Sun's declination, - 21 11 43 N. 

Sum is 63° 19' 22", and difference is 20© 54' 56". 
Howto find From C, make (712, equal to the sine of the difference of 

the points in ^ ^ ^ v 2 q 54, 56 " ) and Q d fa gine f fa^ gum 
the ellipse v ■" 

marking the (63° 19' 22"). 

ViSi he lace Divide ( 12 ) d int0 tW0 e( l Ual P artS at tne P° int 9* and 0n 

over 6 P the^ (12), as radius, mark the sine of 15°, 30°, 45°, 60°, 75°, 
earth's disc.. 90°; the line 7, 5, runs through the first point; 8, 4, through 
the second, &c. 

Subtract the latitude (42° 6' 39") from 90°, thus finding 
the co-latitude (47° 53' 21"). On the semidiameter of the 
earth's disc, as radius, take the sine of the co-latitude (47° 
53'), and set off that distance from g, both ways to 6 ; thus 
making a line, 6, 6, at right angles to the universal meridian, 
Cq. On g (6), as radius, and from the pointy, as a center, 
find the sine of 15°, 30°, 45°, &c, and set off those distances 
each way from g, and through the points, thus found, draw 
lines parallel to g C ; these lines, meeting the lines drawn par- 
allel to 6^6, will define the points 5, 6, 7, 8, &c, to 12, and 
1,-2, 3, &c, to 7, the hours of the day on the elliptic curve. 
That is, our supposed observer at the moon would see Boston 

Explanation ii-i -i • i t» n i 

of the hours ( or an y other place in the same latitude as Boston), at the 
round the el- point 9, when it is 9 o'clock at the place, and at 12, when it 
is noon at the place, &c. 

* The reader is supposed to understand how to draw a sine to any 
arc, corresponding to any radius, either with or without a sector. 



ECLIPSES 




293 

Chap. V. 



294 ASTRONOMY. 

chap. v. As this curve touches the disc before 5, and after 7, it 
shows that, in that latitude, on the day in question, the sun 
will rise before 5 in the morning, and set after 7 in the even- 
ing. If the declination of the sun had been as much south as 
now north, the point, d, would have been 12 at noon, and all 
the hours would have been on the upper part of the ellipse, 
which is not now represented. 

From C, as in the general eclipse, set off the distance, C n, 
equal to the moon's latitude, and, through the point n, draw 
the moon's path at right angles to CL. 

As the ellipse represents the sun's path on the disc, and as 
the point (12) refers, of course, to apparent noon, and not to 
mean noon, therefore, we will mark off the time on the moon's 
path, corresponding to apparent time. 
How to mark When the moon's center passes the point n, it is at ecliptic 
mo v atr> 6 C0n j unc ^ 0n > apparent time, at Boston, or it must be considered 
the apparent time, corresponding to any other meridian for 
which the projection may be intended. 

The ecliptic d , apparent time, Greenwich, is 8h. 49 m. Os. 
For the longitude of Boston, subtract 4 44 16 

Conjunction, apparent time, at Boston, 4 4 44 

The moon's hourly motion from the sun is 27' 39": take 

this distance from the scale, in the dividers, and make the 

small scale, ab, which divide into 60 equal parts, then each 

in this case, P ar * corresponds with a minute of the moon's motion from the 

the ellipse SUI1} aa( j the distance, ab, will correspond with one hour of the 

mence ^be'. moon 's motion along its path. At 4h. 4 m. 44 s. the moon's 

tween 4 and center will be at the point n, the sun's center, at the same 

5 o'clock. t j me ^ w -jj ^ j ugt k e y 0n( j t b e p i nt 4^ on t h e ellipse ; and, as 

the distance between these two points is greater than the sum 
of the semidiameters of sun and moon, therefore, the eclipse 
will not then have commenced; but the moon moves rapidly 
along its path, and, at 5 o'clock, the center of the moon will 
be at the point marked 5 oh the moon's path, and the center 
of the sun will be at the point marked 5 on the ellipse, and 
these two points are manifestly so near each other, that the 
limb of the moon must cover a part of that of the sun, show- 



ECLIPSES. 205 

ing that the eelipse must have commenced prior to that time. Chap. v. 

To find the time of commencement more exactly, let the hour To find the 

on the moon's path be subdivided into 10 or 5-minute spaces, more exaet 

and take the sum of the semidiameter of the sun and moon 

in your dividers from the scale CD, and, with the dividers 

thus open, apply one foot on the moon's path, and the other 

on the sun's path, and so adjust them that each foot will stand 

at the same hour and minute on each path as near as the eye 

can decide. The result in this case is 4h. 28 m. The end of 

the eclipse is decided by the dividers in the same manner, and, 

as near as we can determine, must take place at 6h. 44 m. 

To find the time of greatest obscuration, we must look How t°find 
along the moon's path, and discover, as near as possible, from grea test ob- 
what point a line drawn at right angles from that path, will scuration. 
strike the sun's path at the same hour and minute; the 
time, thus marked on both paths, will be the time of great- 
est obscuration. 

In this case it appears to be 5 h. 40 m., and the two cen- 
ters are very nearly together ; so near, that we cannot decide 
on which side of the sun's center, the moon's center will be, 
without a trigonometrical calculation. 

To show a representation of an eclipse at any time during How to find 
its continuance, we must take the semidiameter of the sun in the ma s ni - 
the dividers trom the scale ; and, from the point or time on eclipse, 
the sun's path, describe the sun ; and, from the same point of 
time on the moon's path, describe a circle with the radius of 
the moon's semidiameter ; the portion of the sun's diameter 
eclipsed, measured by the dividers, and compared with the 
whole diameter, will give the magnitude of the eclipse as near 
as it can be determined by projection. 

The results of this projection are as follows : 

App. time. Mean time. 

Beginning of the eclipse, p. m., 4h. 28 m. 4h. 24 m. 39 s. Accuracy of 
Greatest obscuration, 5 40 5 36 39 theresQlte - 

End of the eclipse, 6 44 6 40 39 

From the projection the two centers are nearer together 
than the difference of the semidiameter of the sun and moon, 
20 



296 ASTRONOMY. 

Chap. v. and the moon's diameter being least, the eclipse will be an- 
nular, as represented in the projection. 

The above results are, probably, to be relied upon to within 
three minutes. 

We have now done with the projection, as far as the particu- 
lar locality, Boston, is concerned ; but, in consequence of the 
facility of solution, we cannot forbear to solve the following 
problem : In the same parallel of latitude as Boston, find the 
longitude where the greatest obscuration will be exactly at 2 p. M., 
apparent time. 
a very easy From the point 2, in the ellipse, draw a line at right an- 
ta t rTiem" &^ es *° ^ e moon's path, and that point must also be 2h. on 
the moon's path; running back to conjunction, we find it 
How solved, must take place at 1 h. 50 m. ; but the conjunction for Green- 
wich time is 8 h. 49 m., the difference is 6 h. 59 m., correspond- 
ing to 104° 45' west longitude ; we further perceive that the 
sun would there be about 9 digits eclipsed on the sun's south- 
ern limb. 
How to find Now, admitting this construction to be on mathematical 

more accu- p r i nc }pi es / ag ft really is, except the variabilis of the de- 
rate results. r r x . . 

ments), we can determine the beginning and end of a local 
eclipse to great accuracy, by the application of analytical 

GEOMETRY. 

enera j^ q jy an( j C 05 be two rectangular co-ordinates, then 

equations to ° 

aid in com- the distance of any point in the projection from the center 
putmgaiithe can j^ determined by means of equations. 

circumstan- 

ces of an Let x and y be the co-ordinates of any point on the sun s 

eclipse as path or elliptic curve, and Xand Y the co-ordinates of any 
en e place, point on the moon's path, then we have the following equa- 
tions : 

( 1 ) y=p sin. L cos. DztP cos - & sm - D cos - i { solar 
( 2 ) x=p cos. L sin. t ) co-ordin. 

( 3 ) Y=d±h i sin. B j lunar co _ ordinateg , 
(4) X=hicos.B ) 

In these remarkable equations, p is the semidiameter of pro- 
jection, L the latitude, D the sun's declination, t the time 
from apparent noon, d the difference in declination between 



ECLIPSES. 297 

sun and moon at the instant of conjunction in right ascen- Chap. v. 
sion, h the moon's hourly motion from the sun, i the interval 
of time from conjunction in right ascension — minus, if before 
conjunction — plus, if after ; and £ is the angle L C <s>, or the 
angle which the moon's path makes with CD. 

In the equations x and X, are horizontal distances. In 
equation ( 1 ) the plus sign is taken when the hours are on 
the upper side of the ellipse, as in winter ; when on the lower 
side take the minus sign. 

In equation ( 3 ), the plus sign is taken when the motion of Explanation 
the moon is northward, and the minus sign, when southward. of the s y m_ 
lhe sin. t, or cos. t, means the sin. or cos. of an arc, corre- 
sponding to the time at the rate of 15° to one hour. 

The solar and lunar co-ordinates, or equations ( 1 ), ( 2 ), The symbol 
( 3 ), and ( 4 ), are connected together by the following equa- d 

expresses tiis 

tions ; the minus sign applies to forenoon, the plus sign to time of con- 
afternoon : junction in 

r £_ 4. right ascen- 

' sion. 

^ -\-iz=t. 

To apply these equations, and, of course, the former ones, 
i, the interval of time from conjunction must be assumed, and, 
as the time of conjunction is known, t thus becomes known ; 
d, h, and B, are known by the elements, therefore, x, y, and 
X, Y, are all known. But the distance between any two 
po'nts referred to co-ordinates, is always expressed by 



J(xmXy+{ycnYy. 

When an eclipse first commences, or just as it ends, this ex- 
pression must be just equal to the semidiameter of the sun 
and moon ; and if, on computing the value of this expression, 
it is found to be less than that quantity, the sun is eclipsed; 
if greater, the sun is not eclipsed ; and the result will show 
how much of the moon's limb is over the sun, or how far 
asunder the limbs are, and will, of course, indicate what 
change in the time must be made to correspond with a con- 
tact, or a particular phase of the eclipse. 

For an eclipse absolutely central, and at the time of being 
central, the last expression must equal zero; and, in that 



298 



ASTRONOMY. 



chap. v. case, x=X, and y= Y. In cases of annular eclipses, to find 
the time of formation or rupture of the ring, the expression 
must be put equal to the difference of the semidiameters of 
sun and moon. In short, these expressions accurately, ef- 
ficiently, and briefly cover the whole subject; and we now 
close by showing their application to the case before us 

By the projection we decided that the beginning of the 
eclipse would be at 4h. 28 m., apparent time at Boston. Call 
this the assumed or approximate time, and for this instant we 
will compute the exact distance between the center of the sun 
and the center of the moon, and if that distance is equal to 
the sum of their semidiameter, then 4h. 28 m. is, in fact, the 
time, otherwise it is not, &c. 



Application 
of the preced- 
ing expres- 
sions. 



An accurate 
computation 
for the begin- 
ning of the 
eclipse as 
seen from 
Boston. 



h. m. s. 

4 13 21 
15 



Conjunc. in R. A., app. time, Boston, 
Assume i equal to, 

Therefore, t is equal to 4 28 21=67° 5' 15". 

^=54' 39"=3279. Reduced lat., Z=42° 6' 38". 

i>=21° 11'43"; d=Cr=1324".3; £=1659 i= 

B=U° 0' 58". 

p 3279 - log. 3.515741 - log. 3.515741 

42° 6' 38" sin. 9.826437 - cos. 9.870315 

21 11 43 cos. 9.969583 - sin. 9.558149 

67 5 15 cos. 9.590288 



i . 

4 ? 



L 

D 

t 



2050.1 
342.3 

y=T707\8 



log. 3.311761 .342.3 log. 2.534493 



P 

cos. L 
t 



sin. 



3.515741 
9.870315 
9.964303 



*=2240.5 log. 3.350359 



For Fand X: 
B 16° 0' 58" - 
A$414".75 

114.5 
add 1324.3 

r= ~1438.8 
(Fc/> y)=269 



sin. 9.440775 
log. 2.617800 



cos. 9.982804 
log. 2.617800 



2.058575 398.6 2.600604 

X=398.6 
(sc/>X)=1841.9. 



ECLIPSES. 299 

Here are two sides of a right-angled triangle, and the hy- Chap. v. 
pothenuse of that triangle is 1861". 8, which is the distance 
between the center of the sun and moon at that instant ; but 
the semidiameter of the sun and moon is only 1853"; there- The eclipse 
fore the eclipse has not yet commenced, and will not until the ™* 
moon moves over 8". 8; which will require about 19 s., as we 
determined by proportion, because the apparent motion of the 
moon will be almost directly toward the sun. 

When the apparent motion of the moon is not so nearly in 
a line with the sun, as it is in this case, we cannot proportion 
directly to the result of the correction. In fact, the apparent 
motion of the moon is on one side of a plane right-angled tri- 
angle, and the distance between the center of sun and moon 
is the hypothenuse to that triangle, and the variation of the 
moon on its base, varies the hypothenuse, and the computa- 
tion must be made accordingly. 

Hence, to the assumed time of beginning, 4h. 28 m. 21s. 
Add, 19 

Beginning, apparent time, - - 4 28 40 

Mean time, - - - - 4 25 19 

By the application of the same expressions, we learn that The moon ' s 
the greatest obscuration will take place at 4h. 41m., mean ren tiyi8"N. 
time at Boston ; and the apparent distance of the mooii's cen- of the sun ' s 
ter will be 18" north of the sun's center ; and, as the moon's „, . apparen ' 

' ' conjunction. 

semidiameter is 57" less than that of the sun, a ring will be 
formed of between 10" and 11" wide at the narrowest point. 
End of the eclipse, 6h. 46 m. 58 s., mean time. 

In computing for the end of the eclipse, we assumed 
t =sl h. 33 m., and as t is more than 6 h., the second part of 
y changes sign, as we see by the figure; the sun after 6, must 
be above the line 6^6. 

Occultations of stars are computed on the same principles 
as an eclipse of the sun, the star having neither diameter nor 
parallax. 

As problems, to give practice to the learner, we take the 
elements of two solar eclipses for 1846, from the Nautical 
Almanac, with their results, as answers to the problems : 



300 ASTRONOMY. 

Chap. "V. 



Examples 
given for 
practice. 



ELEMENTS OP THE ECLIPSES OF THE SUN. 

1846. April 25. October 19. 

h. m. s. h. m. s. 

Greenwich M. T. of 6 in R. A., 4 55 54 -5 19 50 12 .2 

O and (§'s Right Ascension, 2 11 8-31 13 38 31 -54 

o i ii a i it 

(§'s declination, N. 13 25 19 -8 S. 10 23 43 -0 

O's declination, N. 13 13 21 -2 S.10 15 3-9 

O 's hourly motion in R. A., 33 55 1 30 42 -2 

O's hourly motion in R. A., 2 21 -3 2 21 -5 

®'s hourly motion in dec. N. 8 23-6 S. 8 37*0 

O's hourly motion in dec. N. 48 -8 S. 54*1 

O's equatorial hor. parallax, 57 53 -8 55 33-4 

O's equatorial hor. parallax, 8-5 8-6 

$ 's true semidiameter, 15 46 -5 15 8 -4 

O's true semidiameter, 15 54-5 16 5-6 



THE APRIL ECLIPSE. 

General re- Begins on the earth generally April 25 d. 2h. 2 m. 4 s., mean 

time at Greenwich, in longitude 119° 40' W. of Greenwich, 

and latitude 6° 15' S. 
Central Eclipse begins generally April 25 d. 3 h. 3 m. 3 s. 

in longitude 135° 51' W. of Greenwich, and lat. 2° 11' S. 
Central eclipse at noon, April 25 d. 4 h. 55 m. 9 s. 

in longitude 74° 31' W. of Greenwich, and lat. 25° 21' N. 
Central eclipse ends generally April 25 d. 6 h. 37 m. 6 s. 

in longitude 3° 43' W. of Greenwich, and lat. 24° 56' N. 
Ends on the earth generally April 25 d. 7h. 38 m. 5 s. 

in longitude 20° 4' W. of Greenwich, and lat. 20° 52' N. 

THE OCTOBER ECLIPSE. 

Begins on the earth generally October 19 d. 16h. 46 m. 7 s. 
mean time at Greenwich, in longitude 16° 21' E. of Green- 
wich, and latitude 9° 50' N. 

Central eclipse begins generally October 19 d. 17 h. 52 m. s. 
in longitude 0° 32' W. of Greenwich, and lat. 6° 44'N. 

Central eclipse at noon, October 19 d. 19 h. 50 m. 2 s 

in longitude 58° 41' E. of Greenwich, and lat. 19° 22' S. 



ECLIPSES. 301 

Central Eclipse ends generally October 19 d. 21 h. 38 m. 9 s. Chap. v. 

in longitude 126° 5' E. of Greenwich, and lat. 23° 51' S. 
Ends on the earth generally October 19 d. 22 h. 44 m. 1 s. 

in longitude 109° 6' E. of Greenwich, and lat. 20° 47' S. 

The following is a catalogue of the solar eclipses that will 
be visible in New England and New York, between the years 
1850 and 1900; the dates are given in civil, not astronomi- 
cal, time. 

1851, July 28th. Digits eclipsed, 3f, on sun's northern limb. statistics 

of* ©cllDS6S 

1854, May 26th. As computed in the work. from 1850 10 

1858, March 15th. Sun rises eclipsed. Greatest obscura- woo. 

tion, 5^ digits on sun's southern limb. 

1859, July 29th. Digits eclipsed, 2|, on sun's northern limb. 

1860, July 18th. Digits eclipsed, 6, on sun's northern limb. 

1861, December 31st. Sun rises eclipsed. Digits eclipsed 

at greatest obscuration, 4±-, on sun's southern limb. 

1865, October 19th. Digits eclipsed, 8^, on sun's southern 

limb. 

1866, October 8th. ± digit eclipsed. South of New York 

no eclipse. 
1869, August 7th. Digits eclipsed, 10, on sun's southern 

limb. This eclipse will be total in North Carolina. 
1873, May 25th. Sun and moon in contact at sunrise, 

Boston. 

1875, September 29th. Sun rises eclipsed. This eclipse 

will be annular in Boston, Maine, New Hampshire, 
and Vermont. 

1876, March 25th. Digits eclipsed, 3^, on sun's northern 

limb. 
1878, July 29th. Digits eclipsed, 7^, on sun's southern 

limb. This is the fourth return of the total eclipse 

of 1806. 
1880, December 31st. Sun rises eclipsed. Digits eclipsed 

at greatest obscuration, 5^, on sun's northern limb. 
1885, March 16th. Digits eclipsed, 6±, on sun's northern 

limb. 



302 ASTRONOMY. 

Chap - v - 1886, August 28th. North of Massachusetts no eclipse; 
statistics south, sun eclipsed. 

f^ 6 ^]^ 1892, October 20th. Digits eclipsed, 8, on sun's northern 
woo. limb. 

1897, July 29th. Digits eclipsed, 4^, on sun's southern 

limb. 
1900, May 28th. Digits eclipsed, 11, on sun's southern 
limb. The sun will be totally eclipsed in the State 
of Virginia. 



TABLES. 

EXTRACTS FROM THE NAUTICAL ALMANAC FOR JANUARY, 1846. 



■ 

4 

a 
o 

% 

n 

— 
o 

- 
Q 


THE SUN'S 
Apparent 


Logar. 
of the 
Radius 
Vector 
of the 
Earth. 


THE MOON'S 


i 
Longitude. 


Latitude. 


Longitude. 


Latitude. 


Semi- 
diam. 


Hor. 
Paral. 


i 
Noon. 


Noon. 


Noon. 


Noon. 


Noon. 


Noon. 


Noon. 


o / // 


// 




II 


1 II 


/ // 


! 
i n 


1 
2 
3 


280 46 15.3 

281 47 26.1 

282 48 36.5 


N.0.49 
0.45 
0.37 


9.99266 
9.99266 
9.99267 


330 42 13.9 
345 7 12.0 
359 4 55.4 


N.4 54 8.5 
4 24 8.7 
3 39 5.9 


16 21.6 
16 8.3 
15 53.9 


60 2.3! 
59 13.5 i 

58 20.5 


4 
5 

6 


283 49 46.5 

284 50 56.1 

285 52 5.3 


0.27 

0.16 

N.0.03 


9.99267 
9.99268 
9.99268 


12 35 34.7 
25 41 31.5 
38 26 25.0 


2 43 1.9 

1 39 55.7 

N.O 33 28.3 


15 39.8 
15 26.7 
15 15.2 


57 28.7 
56 40.8 ! 
55 58.71 


7 
8 
9 


286 53 13.9 

287 54 22.0 

288 55 29.7 


S.0.11 

0.25 
0.38 


9.99270 
9.99271 
9.99272 


50 54 23.2 
63 9 30.1 
75 15 21.8 


8.0 33 3.6 

1 36 46.8 

2 35 8.6 


15 5.6 
14 57.6 
14 51.5 


55 23.31 
54 54.1 1 
54 31.6 


10 
11 
12 


289 56 36.8 

290 57 43.4 

291 58 49-5 


0.49 
0.58 
0.65 


9.99274 
9.99277 
9.99279 


87 14 56.3 

99 10 31.3 

111 3 50.8 


3 25 55.4 

4 7 13.7 
4 37 30.7 


14 46.9 
14 43.8 
14 42.1 


54 14.6 
54 3.3 
53 57.0 


13 
14 
15 


292 59 55.3 

294 1 0.5 

295 2 5.4 


0.70 

0.71 
0.69 


9.99282 
9.99285 
9.99288 


122 56 17.6 

134 49 7.9 
146 43 48.4 


4 55 38.9 

5 56.4 
4 53 7.6 


14 41.7 

14 42.8 
14 45.5 


53 55.7 

53 59.8 

54 9.7 


16 
17 

18 


296 3 9.9 

297 4 14.0 

298 5 17.8 


0.64 
0.57 
0.47 


9.99292 
9.99295 
9.99299 


158 42 11.3 

170 46 44.8 
183 38.7 


4 32 23.1 
3 59 17.1 
3 14 47.1 


14 50.0 

14 56.3 

15 4.6 


54 26.0 

54 49.0 

55 19.7 


19 
20 
21 


299 6 21.2 

300 7 24.2 

301 8 26.7 


0.35 
0.23 

1 S.0.09 

i 


9.99304 
9.99308 
9.99313 


195 27 41.8 
208 12 10.4 
221 18 27 5 


2 20 14.2 

1 17 27.8 

S.O 8 53.1 


15 15.2 

15 27.7 
,15 42.0 


55 58.4 

56 44.4 

57 37.0 


22 
23 
24 


302 9 28.9 

303 10 30.4 

304 11 31.3 


N.0.04 
0.15 
0.25 


9.99318 
9.99323 
9.99328 


234 50 26.7 
248 50 42.5 
263 19 30.4 


N.l 2 20.5 

2 12 11.7 

3 15 50.9 


15 57.3 

16 12.5 
16 26.2 


58 32.9 

J59 28.8 
60 19.0 


25 

26 

27 


305 12 31.5 

306 13 30.9 

307 14 29.3 


0.33 

0.38 

0.40 
i 


9.99334 
9.99339 
9.99345 


278 13 48.8 
293 26 49.2 

308 48 22.8 


4 8 2.8 
4 43 49.4 
4 59 32.4 


Il6 36.8 
16 42.9 
16 43.5 


60 57.9 

61 20.2 
61 22.6 


28 
29 
30 
31 


308 15 26.8 

309 16 23.3 

310 17 18.5 

311 18 12.6 


0.40 
| 0.37 

0.30 
I 0.21 


9.99351 
9.99357 
9.99363 
9.99369 


324 6 34.0 

339 9 55.3 

353 49 32.0 

8 13.1 


4 53 45.4 
4 27 32.9 
3 44 8.2 

2 47 58.7 


16 38.7 
16 28.9 
16 15.6 
16 0.2 


61 4.9 
60 29.1 
59 40.2 
58 43.7 


32 


1312 19 5.3 


1 N.0.10 19.99375 


21 40 34.3 


N.l 43 50.6 


15 44.2 


57 45.1 



TABLES. 



TABLE I. 

MEAN ASTRONOMICAL REFRACTIONS. 
Barometer 30 in. Thermometer, Fah. 50°, 



Ap. Alt, 
"IPO 7 " 


Refr. 


Ap. Alt. 


Eefr. 


Ap. Alt. 


Refr. 


Alt. 


Refr. 


33' 51" 


4° 0' 


11' 52" 


12° 0' 


4' 28.1" 


42° 


1 4.6' 


5 


32 53 


10 


11 30 


10 


4 24.4 


43 


1 2.4 


10 


31 58 


20 


11 10 


20 


4 20.8 


44 


1 0.3 


1 15 


31 5 


30 


10 50 


30 


4 17.3 


45 


58.1 


20 


30 13 


40 


10 32 


40 


4 13.9 


46 


56.1 


25 


29 24 


50 


10 15 


50 


4 10.7 


47 


54.2 


30 


28 37 


5 


9 58 


13 


4 7.5 


48 


52.3 


35 


27 51 


10 


9 42 


10 


4 4.4 


49 


50.5 


40 


27 6 


20 


9 27 


20 


4 14 


50 


48.8 


45 


26 24 


30 


9 11 


30 


3 58.4 


51 


47.1 ' 


50 


25 43 


40 


8 58 


40 


3 55.5 


52 


45.4 


55 


25 3 


50 


8 45 


50 


3 52.6 


53 


43.8 


1 


24 25 


6 


8 32 


14 


3 49.9 54 


42.2 


5 


23 48 


10 


8 20 


10 


3 47.1 


55 


40.8 


| 10 


23 13 


20 


8 9 


20 


3 44.4 


56 


39.3 


15 


22 40 


30 


7 58 


30 


3 41.8 


57 


37.8 


20 


22 8 


40 


7 47 


40 


3 39.2 


58 


36.4 


25 


21 37 


50 


7 37 


50 


3 36.7 


59 


35.0 


30 


21 7 


7 


7 27 


15 O 


3 34.3 


60 


33.6 


35 


20 38 


10 


7 17 


15 30 


3 27.3 


61 


32.3 


40 


20 10 


20 


7 8 


16 


3 20.6 


62 


31.0 


45 


19 43 


30 


6 59 


16 30 


3 14.4 


63 


29.7 


50 


19 17 


40 


6 51 


17 


3 8.5 


64 


28.4 


55 


18 52 


50 


6 43 


17 30 


3 2.9 


65 


27.2 


2 


18 29 


8 


6 35 


18 


2 57.6 


66 


25.9 


5 


18 5 


10 


6 28 


19 


2 47.7 


67 


24.7 


10 


17 43 


20 


6 21 


20 


2 38.7 


68 


23.5 


15 


17 21 


30 


6 14 


21 


2 30.5 


69 


22.4 


20 


17 


40 


6 7 


22 


2 23.2 


70 


21.2 


25 


16 40 


50 


6 


23 


2 16.5 


71 


19.9 


30 


16 21 


9 


5 54 


24 


2 10.1 


72 


18.8 


35 


16 2 


10 


5 47 


25 


2 4.2 


73 


17.7 


40 


15 43 


20 


5 41 


26 


1 58.8 


74 


16.6 


45 


15 25 


30 


5 36 


27 


1 53.8 


75 


15.5 I 


50 


15 8 


40 


5 30 


28 


1 49.1 


76 


14.4 j 


55 


14 51 


50 


5 25 


29 


1 44.7 


77 


13.4 ! 


3 


14 35 


10 


5 20 


30 


1 40.5 


78 


12.3 ! 


5 


14 19 


10 


5 15 


31 


1 36.6 


79 


11.2 


10 


14 4 


20 


5 10 


32 


1 33.0 


80 


10.2 


15 


13 50 


30 


5 5 


33 


1 29.5 


81 


9.2 


20 


13 35 


40 


5 


34 


1 26.1 


82 


8.2 


25 


13 21 


50 


4 56 


35 


1 23.0 


83 


7.1 j 


30 


13 7 


11 


4 51 


36 


1 20.0 


84 


6.1 


35 


12 53 


10 


4 47 


37 


1 17.1 


85 


5.1 


40 


12 41 


20 


4 43 


38 


1 14.4 


86 


4.1 


45 


12 28 


30 


4 39 


39 


1 11.8 


87 


3.1 


50 


12 16 


40 


4 35 


40 


1 9.3 


88 


2.0 


55 


12 3 


50 


4 31 


41 


1 6.9 


89 


1.0 



TABLE 



CORRECTION OF MEAN REFRACTION. 



Plight of the Thermometer. 



i App. 1 24 c 
Alt. i 
o t '+" 


28° j 32° 


36° 


40° 


440 


52° 


56° 


60° 


64° 


68° 


72C 


76° 80°i 


'+" 


'+" 


tit' 


'+" 


+" 


1 // 


1 n 


/ // 


r /; 


/ // 


/ _ / 


1 //| /_ // 


0.002.18 


1.55 


1.33 


1.11 


51 


31 


10 


29 


48 


1.07 


1.25 


143 


2.01|2.19| 


0.10|2.12 


1.49i 1.28 


1.08 


48 


29 


9 


27 


45 


1.04 


1.21 


1.38 


1.542.12 


0.20:2.05 


1.44J 1.24 


1.04 


46 


28 


9 


26 


44 


1.01 


1.17 


1.33! 


1.4912.05 


0.301.59 


1.39 1.20 


1.01 


44 


26 


8 


25 


41 


58 


1.13 


1.28 


1.4311.591 


0.40b 53 


1.34 


1.16 


58 


42 


25 


8 


24 


39 


55 


1.10 


1.24 


1.38 


1.53 : 


0.5011.48 


1.29 


1.12 


55 


40 


24 


8 


23 


37 


52 


1.06 


1.20! 


1.34 


1.48 


1.00J1. 43 


1.25 


1.09 


53 


38 


23 


7 


21 


36 


50 


1.03 


1.17 


1.30 


1.43 


1.10I1.38 


1.21 


1.06 


50 


36 


22 


7 


20 


34 


48 


1.00 


1.13 


1,26 


1.38 


1.201.33 


1.17 


1.03 


48 


34 


21 


6 


19 


32 


45 


57 


1.09 


1,21 


1.33 


1.3011.29 


1.14 


1.00 


46 


32 


20 


6 


18 


31 


43 


54 


1.06 


1.18 


1.29 


1.401.25 


1.11 


57 


44 


31 


18 


6 


18 


30 


41 


52 


1.04 


1.15 


1.25 


1.50I1.21 


1.08 


55 


42 ! 


30 


17 


6 


17 


28 


39 


50 


1.01 


1.11 


1.21 


2.00!i.i8 


1.05 


53 


39 


29 


17 


5 


16 


27 


37 


48 


56| 


1.08 


1.18 


2.20 1.11 


1.00 


48 


37 


26 


16 


5 


15 


25 


35 


44 


54 


1.03 


1.11 


2.40'1.06 


55 


44 


34 


24 


14 


5 


14 


23 


32 


41 


50 


58 


1.06 


S.OOjl.Ol 


51 


41 


32l 


22 


13 


4 


13 


21 


30 


38 


46 


54 


1.01 


3.201 57 


47 


38 


29J 


21 


13 


4 


12 


20 


28 


35 


43 


50 


.57 


3.40l 53 


44 


36 


28! 20 


12 


4 


11 


18 


26 


33 


40 


47 


53 


4.00J 49 


41 


33 


26; 18 


11 


4 


10 


17 


24 


31 


37 


44 


50 


4.30J 45 
5.001 41 


38 


31 


24! 17 


10 


3 


9 


16 


22 


28 


34 


40 


45 


35 


28 


221 16 


9 


3 


9 


14 


20 


26 


31 


36 


40 


5.30; 38 


32 


26 


20; 14 


9 


3 


8 


13 


19 


24 


29 


34 


38 


6.00 


35 


30 


24 


19 


13 


8 


2 


7* 


12 


17 


22 


26 


31 


35 


6.30 


33 


28 


22 


17 


12 


7 


2 


7 


11 


15 


20 


24 


29 


33 


7.00 


31 


26 


21 


16 12 


7 


2 


6' 


10 


14 


19 


23 


27 


31 


8 


27 


23 


19 


15! 10 


6 


2 


5 


9 


13 


16 


20 


24 


27 


9 


24 


20 


16 


13i 9 


5 


2 


5 


8 


11 


14 


18 


21 


24! 


10 


22 


18 


15 


12! 8 


5 


1 


4 


7 


10 


13 


16 


19 


22 i 


11 


20 


17 


14 


11 


8 


5 


1 


4 


7 


9 


12 


15 


18 


20 


12 


18 


15 


13 


10 


7 


4 


1 


4 


6 


9 


11 


13 


16 


18 


1.3 


17 


14 


12 


9 


7 


4 


1 


3 


6 


8 


10 


12 


15 


17 


14 


16 


13 


11 


8 


6 


4 


1 


3 


5 


7 


9 


11 


14 


16 


15 


15 


12 


10 


8 


6 


3 


1 


3 


5 


7 


9 


11 


13 


15 


16 


14 


12 


9 


7 


5 


3 


1 


3 


5 


6 


8 


10 


12 


14 


17 


13 


11 


9 


7 


5 


3 


1 


3 


4 


6 


8 


9 


11 


13 


18 


12 


10 


8 


6 


5 


3 


1 


2 


4 


6 


7 


9 


10 


12 


19 


11 


9 


8 


6 


4 


3 


1 


2 


4 


5 


7 


8 


10 


11, 


20 


11 


9 


7 


6 


4 


2 


1 


2 


4 


5 


6 


8 


9 


Hi 


21 


10 


9 


7 


5 


4 


2 


1 


2 


3 


5 


6 


7 


9 


10 


22 


10 


8 


7 


5 


4 


2 


1 


2 


3 


5 


6 


7 


8 


10 


23 


9 


8 


6 


5 


4 


2 


1 


2 


3 


4 


6 


7 


8 


9 


24 


9 


7 


6 


5 


3 


2 


1 


2 


3 


4 


5 


6 


8 


9 


25 


t 


7 


6 


5 


3 


2 


1 


2 


3 


4 


5 


6 


7 


8 


26 


£ 


7 


6 


4 


3 


2 


1 


2 


3 


4 


5 


6 


7 


8 


27 


£ 


6 


5 


4 


3 


2 


1 


2 


3 


4 


5 


6 


7 


8 


2S 


7 


6 


5 


4 


3 


2 





1 


2 


3 


5 


5 


6 


7 


30 


, 7 


6 


5 


4 


3 




2 




29.75 


1 
+ 


2 
4- 


3 

+ 


4 

+ 


5 


6 


7 




— 










28.26 


J28.56 


28.85 


29.15 


30 05)30.35 


30.64 


30.92 








Hi 


ght oi 


' the Barometer. 









20 



TABLES. 
TABLE II. 

MEAN PLACES EOR 100 PRINCIPAL FIXED STARS, FOR JAN. 1, 1846. 



Star's Name. 



et Andromeda, 

y Pegasi (Algenib),. . . 

Hydri, 

at Cassiope^e, 

£ Ceti, 

at Urs.Min. (Polaris),. 
fliCeti, 

at Eridani (Achernar), . 

<* Arietes, 

y Ceti, 

at Ceti, 

at Persei, 

» Tauri, 

yi Eridani, 

at Tauri (Aldebarari),. . 
at Auriga (Capella),. . . 

$ Orionis (Rigel), ... . 

Tauri 

J" Orionis, 

at Lepris, 

i Orionis, 

«■ Columbae, 

i* Orionis, 

[a Geminorum, 

* Argus (Canopus), . . . 

51 (Hev.) Cephei, 

at Canis Maj. (Sirius),. 
s Canis Majoris, 

<f Geminorum, 

at 2 Geminor. (Castor),.. 
at Can. Min. (Procyon), 
Geminor. (Pollux),.. 

15 Argus, 

i Hydrae, 

/ Ursse Majoris, 

t Argus, 

at HYDRiE, 

Ursae Majoris, 

c Leonis, 

a Leonis (Regulus), . . , 



1 

2.3 
3 

3 

2.3 
2.3 

3 

1 

3 
3 

2.3 
2.3 

3 

2.3 

1 

1 

I 

2 

2 

3.41 



2.3 

2 
1 
3 

1 

6 

1 

2.3 

3.4 

3 
1.2 

2 

3. 

4 

3 

2 



4 



Right Ascen. 



h. m. s. 

26.257 

5 18.691 

17 34.168 

31 48.294 

35 51.339 

1 3 52.226 
1 16 19.692 
1 31 58.291 

1 58 30.193 

2 35 19.633 

2 54 14.072 

3 13 21.403 

3 38 20.382 

3 50 50.760 

4 27 5.404 

5 5 19.317 

5 7 8.383 

5 16 33.662 

5 24 8.428 

5 25 56.406 

5 28 24.062 
5 34 4.531 

5 46 50.189 

6 13 38.621 

6 20 32.145 
6 26 30.287 
6 38 21.883 

6 52 34.440 

7 10 55.298 
7 24 46.065 
7 31 14.237 

7 35 53.153 

8 59.232 
8 38 37.154 

8 48 38.088 

9 12 58.192 

9 20 1.170 

9 22 31.453 

9 37 6.098 

10 10.062 



Annual Var, 



+ 



3.0720 
3.0784 
3.3054* 
3.3418 



+ 2.9995 
17.1346* 
3.0015 
2.2339 

+ 3.3475 

3.1085 
3.1266 
4.2324 

+ 3.5473 

2.7898 
3.4274 
4.4082 

+ 2.8787 
3.7827 
3.0609 
2.6425 

-f 3.0404 
2.1691 
3.2433 
3.6257 

+ 1.3279 
30.7946 
2.6459* 
2.3558 

+ 3.5918 
3.8561 
3.1445* 
3.6829* 

4- 2.5596 
3.1966 
4.1261* 
1.6100 

4- 2.9499 
4.0504* 
3.4258 

4- 3.2211 



Declination. 



deg. min. sec. 

N.28 14 25.40 
N.14 19 37.80 
S.78 7 24.40 
N.55 41 31.08 

S. 18 49 59,01 
N.88 *!9 17.88 
S. 8 58 45.93 
S.58 1 14.34 

N.22 43 53.86 
N. 2 35 1.17 
N. 3 28 55.70 
N.49 18 28.20 

N.23 37 27.73 
S. 13 57 1.50 
N.16 11 41.39 
N.45 50 6.56 

S, 8 23 3.33 

N.28 28 17.49 

S 25 4.86 

S. 17 56 12.77 

S3. 1 18 17.53 
S. 34 9 36.95 
N. 7 22 22.32 
N.22 35 13.16 

S.52 36 49.17 
N.87 15 31.20 
S. 16 30 32.83 
S. 28 45 59.38 



sec. 

420.055 
20.050 
19.997 
19.862 

+19.810 
19.279 
18.952 
18.461 

417.432 
15.621 
14.532 
13.329 

411.620 

10.711 

7.907 

4.737 



N.22 
N.32 
N. 5 
N.28 

S.23 
N. 6 
N.48 
S.58 



15 37.47 

13 12.93 

36 54.95 
23 34.06 

51 50.94 
58 48.51 
38 32.35 

37 49.78 



S. 7 59 39.05 
N.52 22 31.09 
N.24 28 49.46 
N.12 43 2.96 



Ann. Var. 



+ 



4.583 
3.776 
3.123 
2.968 



4- 2.754 

' 2.262 

4 1.149 

— 1.196 

— 1.796 
2,337 

4.484* 
4.562 

— 6.110 
7.253 

8.758* 
8.152 

—10.104 
12.800 
13.464 
14.961 

—15.366 
16.108* 
16.283 

—17.377 



TABLE II. 



Star's Name. 



t, Argus, 

a UaSiE Majoris, 

J 1 Leonis, 

<f Hydrae et Crateris, . 



]g Leonis, 

y Urs^e Majoris, 
JS Charaaeleontis,. 
a » Crucis, 



Corvi, 

12 Canum Venaticorum, 

a Virginis (Spica),. . . . 
» Urs,e Majoris, 



» Bootis, 

Centauri, 

a Bootis, (Arcturus), 
a 2 Centauri, 



g Bootis, 

a2 Librae, 

/g Urs,e Minoris, 
$ Libras, 



A CORON^E BOREALIS, . 

a Serpentis, 

£ Ursse Minoris, 

/giScorpii, 



J" Ophiuchi, 

a Scorpii, (Antares), . . 

» Draconis, 

a Trianguli Australis, . 



s UrsEe Minoris, 
a Herculis, .... 
a- Octantis, 

,t -L RACONIS, .... 



a Ophiuchi, 

y Draconis, 

^iSagittarii, 

$ Urs^e Minoris, 

a Lyr^e (Vega), . 
Lyr^e, 

£ AcQUILiE, 

} AcQUILjE, 



y AcQUILjE, 

a Acq,uil,e, (Altair), . . 

/3 Acq.uii^e, 

«.2Capricorni, 



2 

1.2 

3 

3.4 

2.3 
2 

5 

1 

2.3 

2.3 

1 

2.3 

3 
1 
1 
1 

3 
3 
3 

2.3 

2 

2.3 

4 

2 

3 
1 
3 

2 

4 

3.41 

6 

2 

2 
2 

3.4| 
3 

1 

3 

3 

3.4 

3 

1.2 
3.4 

3 



Right Ascen. 



h. m. s. 

10 39 6.223 

10 54 10.737 

11 5 54.583 
11 11 38.718 

11 41 12.066 

11 45 42.219 

12 9 26.893 
12 18 4.916 

12 26 18.465 

12 48 49.007 

13 17 5.233 
13 41 27.894 

13 47 21.140 

13 53 0.800 

14 8 38.366 
14 29 11.925 

14 38 15.706 
14 42 22.132 

14 51 13.199 

15 8 43.595 

15 28 10.083 
15 36 41.077 
15 49 41.194 

15 56 29.397 

16 6 16.830 
16 19 58.461 
16 21 55.119 
16 32 25 



Annual Var 



Declination. 



090 + 



17 1 55.988 

17 7 37.617 

17 22 55.004 

17 26 57.473 

17 27 47.219 

17 53 1.955 

18 4 33.276 
18 22 0.703 

18 31 43.386 
18 44 23.696 

18 58 19.965 

19 17 43.889 

19 38 56.278 
19 43 16.128 

19 47 44.866 

20 9 30.316 



+ 2.3051 
3.8001 
3.1928 
3.0010 

+ 3.0654* 
3.1874 
3.3409 
3.2710 

+ 3.1342 
2.8403 
3.1512 
2.3525* 

+ 2.8606 
4.1508 
2.7336* 
4.0165* 

-f 2.6229 
+ 3.3102 

— 0.2692 
+ 3.2226 

+ 2.5279 
+ 2.9391 

— 2.3520 
+ 3.4742 

4- 3.1382 

3.6638 
0.7960 
6.2587 



— 6.5328* 
+ 2.7320 
106.8627 
1.3513 

+ 2.7727 
1.3900 
-f 3.5861 
19.2683* 

4- 2.0118 
2.2124 
2.7566 

4- 3.0086 

4- 2.8511 
2.9254* 
2.9446 
3.3315 



deg. min. sec 

S. 58 52 34.26 
N.62 34 51.81 
N.21 21 59.86 
S. 13 56 46.85 

N.15 25 58.12 
N.54 33 3.18 
S. 78 27 26.15 
S. 62 14 39.74 

S. 22 32 39.93 
N.39 9 4.18 
S. 10 21 20.80 
N.50 5 1.45 

N.19 10 21.03 
S. 59 37 33.93 
N.19 59 12.07 
S. 60 11 37.00 

N.27 43 35.23 
S. 15 23 53.52 
N.74 47 5.5 
S. 8 48 38.53 

N.27 14 11.07 
N. 6 54 49. 
N.78 15 55.43 
S. 19 22 44.18 

S. 3 17 35.67 
S. 26 5 4.58 
N.61 51 50.58 
S. 68 44 4.75 

N.82 16 52.30 

N.14 34 12.67 

S. 89 16 10.25 

N.52 25 3.28 

N.12 40 37.11 
N.51 30 33.50 
S.21 5 36.14 
N.86 35 42.58 



Ann. Var. 



N.38 38 35.33 
N.33 11 14.80 
N.13 38 20.49 
N. 2 48 43.64 

N.10 14 31.50 
N. 8 27 54.32 
N. 6 1 33.90 
S. 13 1 4.19 



—18.82 
19.24 
19.50 
19.61 

—19.99 

20.02 
20.04 
19.99 

—19.92 
19.60 
18.94 
18.12 

—17.89 
17.67 
18.94* 
15.12* 

—15.46 
15.23 
14.71 
13.63 

—12.33 
11.74 

10.80 
10.29 

— 9.55 

8.48 
8.32 
7.48 

— 5.03 
4.54 
3,14 

2.88 

— 2.81 

— 0.61 
4- 0.40 
4- 1.91 



4- 2.77 
3.S6 
5.05 

4- 6.67 

+ 8.39 
8.74 
8.55* 
10.74 



TABLES. 



Star's Name. 



tea I I I 

g i Ri^ht Ascen. Annual Var. 



h. m. s. 



a Pavonis, j 2 j20 

^ Ursse Minoris, I 5 20 

* Cygnj, 1 20 

GPCygni, 5.620 



Cygni,. . 
ct Cephei, . 
Aquarii, 
(6 Cephei, . 



i Pegasi,.. 
ct Aquarii, 
a. Gruis, . . 

£ 1 J < gasi, . 



a. Pis. Aus. (Fomalhaut), 
a. Pegasi (Markab), 

i Piscium, 

y Cephei 



3 
3 
3 
3 

2.3 
3 

2 
3 

1 
2 

4.5 
3 



21 
21 
21 

21 



13 25.814 
16 31.309 
36 11.005 
59 59.947 

6 23.073 

14 53.940 
23 26.875 
26 39.120 



Declination. 



Ann. 



deg. min. sec. 

+ 4.8046 IS. 57 13 19.50 

—52.1273 N.88 50 53.54 

+ 2.0418 N.44 43 57.43 

2.6908* N.37 59 42.08 



21 36 37.346 
21 57 52.326 

21 58 29.837 

22 33 46.976 

22 49 7.531 

22 57 5.584 

23 32 1.736 
23 33 4.581 



+ 2.5486 
1.4163 
3.1628 
0.8059 

-1- 2.9441 
3. to. 1 
3.1 4 

2.9s;7 



N.29 35 53.03 
N.61 56 4.55 
S. 6 14 44.46 
N.69 53 7.21 

N. 9 10 17.35 

S. 1 3 56.72 

S.47 42 12.42 

N.10 1 44.67 



+11.03 j 
11.22 
12.64 ! 

17.48*! 

+14.57 
15.07 
15.56 
15.73 

+16.26 
17,28 
17.30 
18.65 



S.30 26 



I 3.3095 

2^776 N.14 22 40.12 

3.0569 IN. 4 47 30.74 

+ 2.4042 lN.76 46 22.01 



12.28+19.11 
19.31 
19.36* 
+19.92 



Those Annual Variations which include proper motion are distinguished by 

an Asterisk. 



sun's right ascension for 1846. 



Day 














of 
Mo. 


January. 


February. 


March. 


April. 


May. 


June. 




h. m. s. 


h. m. s. 


h, m. s. 


h. m. s. 


h. m. s. 


h. m. s. 


1 


18 46 52 


20 59 11 


22 48 17 


41 52 


2 23 6 


4 35 48 


5 


19 4 30 


21 15 22 


23 3 12 


56 26 


2 48 25 


4 52 12 


10 


19 26 21 


21 35 18 


23 21 40 


1 14 43 


3 7 47 


5 12 50 


15 


19 47 57 


21 54 54 


23 40 


1 33 6 


3 27 24 


5 33 34 


20 


20 9 17 


22 14 12 


23 58 14 


1 51 38 


3 47 15 


5 54 22 


25 


20 30 19 


22 33 14 


16 25 


2 10 22 


4 7 20 


6 15 10 


30 


20 51 




34 36 


2 29 17 


4 27 : J .8 


6 35 55 


Day 

of 

Mo. 


July. 


August. 
6 


September. 


October. 


November. 


December. 


h. m. s. 


h. m. s. 


h. m. s. 


h. m. s. 


h. m. s. 


h. m. s. 


1 


6 40 4 


8 44 55 


10 41 


12 29 4 


14 25 16 


16 29 1 


5 


6 56 34 


9 23 


10 55 29 


12 43 36 


14 41 2 


16 46 23 


10 


7 17 5 


9 19 29 


11 13 30 


13 1 54 


15 1 5 


17 8 17 


15 


7 37 25 


9 38 21 


11 31 28 


13 20 24 


15 21 28 


17 30 22 


20 


i 7 57 33 


9 56 60 


11 49 25 


13 39 8 


15 42 14 


17 52 33 


25 


8 17 28 


10 15 27 


12 7 24 


13 58 9 


16 3 19 


18 14 46 


30 


j 8 37 7 


10 33 44 


12 25 27 


14 17 27 


16 24 43 


18 36 57 



The R. A. in this title will answer for corresponding days in other years within 
four minutes, and for periods of four years the difference is only about seven 
seconds for each period. 



TABLE III. 



TABULAR VIEW OF THE SOLAR SYSTEM. 



Names. 



bun 

Mercury- 
Venus 
The Earth 
Mars 
Vesta 
Iris ) 
Hebe ( 
Flora f * 
Astrea / 
Juno 
Ceres 
Pallas 
Jupiter 
Saturn 
Uranus 
Neptune 



Mean diameter 
in miles. 



883000 
3224 
7687 
7912 
4189 
238 

Unknown 

1420 
Not well (160 
known. )120 
89170 
79040 
35000 
35000 



Mean distance 

from the Sun 

in miles. 



37 million 

68 

95 
144 

224,340,000 
226 million 
230 

240 " 
246 
253,600,000 
263,236,000 
265 million 
490 

900 » 
1800 " 
2850 " 



Mean dist.; 
the Earth's 
dist. unity. 



0.3870989. 

0.7233329 

1.0000000 

1.52369210 

2.36120 

2.37880 

2.42190 

2.52630 

2.5895 

2.66514 

2.76910 

2.77125 

5.202776 

9.538786 

19.182390 

29.59 



Log. of 

mean 
distance. 



Time of revolu- 
tions round 
the Sun. 



587818 
859306 
000000 
182810 
373100 
376384 
384004 
402487 
413211 
.425710 
442334 
442725 
.716212 
979476 
282853 
477121 



Log. of 

times of 

revolution, 



DAYS. 

87.969258 
224.700787 
365.256383 
686.979646 
1324.289 
1327.973 
1375. nearly 
1469.76 
1512. nearly 
1594.721 
1683.064 
1685.162 
4332.584821 
10759.219817 
30686.8208 
60128.14 



1.944324 
2.351610 
2.562598 
2.836942 
3.121991 
3.123190 
3.138303 
3.167300 
3.179547 
3.202700 
3.226086 
3.226610 
3.636738 
4.031718 
4.486953 
4.779076 



TABLE III. 

ELEMENTS OF ORBITS FOR THE EPOCH OF 1850, JANUARY 1, MEAN NOON 

AT GREENWICH. 



Planets. 


Inclination 

of orbits to 

ecliptic. 


Variation 
in 100 
years. 


Long, of the 

ascending 

nodes. 


Variation 
in 100 
years. 


Longitude 

of 
Perihelion. 


Variation 
in 100 
years. 


... 

Mean longi- 
tude at 
epoch. 


Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 


' " 

7 18 
3 23 26 

1 51 6 

7 8 29 
13 2 53 
10 37 17 
34 37 44 

1 18 42 

2 29 29 
46 27 


+18.2 

— 4.6 

— 0.2 
—12. 

—22. 

—15. 

3. 


O ' " 

46 34 40 
75 17 40 

48 20 24 
103 20 47 
170 53 

80 47 56 
172 42 38 

98 55 19 
112 22 54 

73 12 


-j-51 

-j-42 
-f-26 


O ' " 

75 9 47 
129 22 53 
100 22 10 
333 17 57 
254 4 34 

54 18 32 
147 25 41 
121 30 3 3 

11 56 

90 7 
168 14 47 


+ 


1-93- 

-78 

103 

110 

157 


O ' " 

327 17 9 

243 58 4 

100 47 1 

182 9 30 1 

113 28 12 

165 17 38 

1 3 10i 

327 31 24! 

160 21 50! 

13 58 13: 

28 20 22! 

i 












r 57 
-51 
-24 




- 95 
U116 

- 87 



* It is with reluctance that we give these planets a place in the tables. The 
fact of their existence is as yet questionable, and their elements, at present, cannot 
be well known. We give the logarithms in the tables, that the data may be at 
hand to exercise the student on Kepler's third law. 



TABLE III. 

TABULAR VIEW OF THE SOLAR SYSTEM. 



Names. Mass. 


Density. 


Gravity. , 


Sidereal 
Rotation. 


Light and 
Heat. 


Mercury. . 

Venus 

Earth 

Mars 

Jupiter 
Saturn. . . . 
Uranus . . . 
Sun 

Moon 

■ 


i 

2 2 5 8 10 

1 

4 12 11 

1 
3 5 5 

1 

2680337 

1 

10 4 8.7 

1 

3 5 0.2 

1 

17 9 18 

1 

1 

26620200 


3.244 
0.994 
1.000 
0.973 
0.232 
0.132 
0.246 
0.256 
0.665 


1.22 
0.96 
1.00 
0.50 
2.70 
1.25 
1.06 
28.19 
0.18 


h. m. s. 
24 5 28 

23 21 7 

24 

24 39 21 
9 55 50 

10 29 17 
Unknown. 

25 12 
27 7 43 


6.680 
1.911 
1.000 
.431 
.037 
.011 
.003 



TABLE III. 



Planets. 


Eccentricities 
of orbits. 


Variation in 100 
years. 


Motion in mean 

long, in 1 year 

of 365 days. 


Mean Daily 
Motion in 
longitude. 


Mercury. . . . 

Earth 

Mars 

Juno 


0.20551494 
0.00686074 
0.01678357 
0.09330700 
0.08856000 
0.25556000 
0.07673780 
0.24199800 
0.04816210 
0.05615050 
0.04661080 


+ .000003868 

— .000062711 

— .000041630 
-f .000090176 
-j- .000004009 

— .000005830 


O / 

53 43 3.6 
224 47 29.7 
—0 14 19.5 
191 17 9.1 


' " 

4 5 32.6 

1 36 7.8 
59 8.3 
31 26.7 
16 17.9 
13 33.7 
12 49.4 
12 48.7 
4 59.3 
0.6 
42.4 








-f .000159350 

— .000312402 

— .000025072 


30 20 31.9 

12 13 36.1 

4 17 45.1 



TABLE III. 



SATELLITES OF JUPITER. 



Satel. 


Mean Distance. 


Sidereal Revolu. 


Inclination of 

orbits to that 

of Jupiter. 


Mass; that 
of Jupiter 

being 
1000000000 


i 

2 
3 
4 


6.04853 

9.62347 

15.35024 

26.99835 


d. 
1 
3 
7 

16 


h. m. 
18 28 
13 14 
3 43 
16 32 


O ' " 

3 5 30 

Variable. 

Variable. 

2 58 48 


17328 
23235 

88497 
42659 



TABLE IV. 



SUN S EPOCHS. 



Years. 




M. 


Long. 


Long. Perigee. 1 


I. 


II. 


III. 


N. 




s. 


o 


' 


a 


s. 


o 


/ 


- ! 




... . . 






1846 


9 


8 


45 


8 


9 


8 


17 


17 


124 


673 


897 


379 


1847 


9 


8 


30 


48 


9 


8 


18 


19 


484 


588 


623 


433 


1848 B. 


9 


9 


15 


37 


9 


8 


19 


20 


878 


505 


151 


487 


1849 


9 


9 


1 


17 


9 


8 


20 


22 


238 


420 


775 


540 


1850 


9 


8 


46 


58 


9 


8 


21 


23 


598 


336 


400 


594 


1851 


9 


8 


32 


39 


9 


8 


22 


24 


958 


250 


025 


648 


1852 B. 


9 


9 


17 


27 


9 


8 


23 


26 


353 


168 


653 


701 


1853 


9 


9 


3 


8 


9 


8 


24 


27 


713 


083 


277 


755 


1854 


9 


8 


48 


48 


9 


8 


25 


29 


073 


998 


902 


809 


1855 


9 


8 


34 


29 


9 


8 


26 


30 


433 


913 


527 


863 


1856 B. 


9 


9 


19 


18 


9 


8 


27 


32 


827 


832 


153 


916 


1857 


9 


9 


4 


58 


9 


8 


28 


34 


187 


746 


779 


970 


1858 


9 


8 


50 


39 


9 


8 


29 


35 


547 


661 


404 


024 


1859 


9 


8 


36 


19 


9 


8 


30 


37 


907 


576 


029 


078 


1860 B. 


9 


9 


21 


8 


9 


8 


31 


38 


301 


494 


656 


131 


1861 


9 


9 


6 


49 


9 


8 


32 


39 


661 


409 


281 


185 


1862 


9 


8 


52 


29 


9 


8 


33 


41 


021 


324 


906 


239 


1863 B. 


9 


8 


38 


10 


9 


8 


34 


42 


381 


239 


530 


292 


1864 


9 


9 


22 


58 


9 


8 


35 


44 


775 


157 


157 


346 


1865 


9 


9 


8 


39 


9 


8 


36 


45 


135 


072 


783 


400 


1866 


9 


8 


54 


20 


9 


8 


37 


47 


495 


985 


408 


453 


1867 


9 


8 


40 





9 


8 


38 


49 


855 


902 


033 


507 


1868 B. 


9 


9 


24 


49 


9 


8 


39 


50 


249 


820 


659 


561 


1869 


9 


9 


10 


30 


9 


8 


40 


52 


609 


734 


285 


615 


1870 


9 


8 


56 


10 


9 


8 


41 


53 


969 


649 


910 


668 


1882 

i 


9 


9 


1 


41 


9 


8 


54 


10 


391 


638 


416 


313 


1871 


9 


8 


41 


51 


9 


8 


42 


54 


329 


564 


534 


721 


1872 B. 


9 


9 


26 


39 


9 


8 


43 


56 


723 


481 


161 


774 


1873 


9 


9 


12 


20 


9 


8 


45 


58 


083 


396 


785 


828 


1874 


9 


8 


58 


1 


9 


8 


47 





443 


311 


410 


881 


1875 


9 


8 


43 


41 


9 


8 


48 


2 


803 


226 


034 


935 


1876 B. 


9 


9 


28 


30 


9 


8 


49 


4 


297 


143 


661 


989 


1877 


9 


9 


14 


10 


9 


8 


50 


5 


657 


058 


286 


042 


1878 


9 


8 


59 


51 


9 


8 


51 


6 


017 


974 


912 


096 


1879 


9 


8 


45 


32 


9 


8 


52 


7 


377 


889 


537 


150 


1880 B. 


9 


9 


30 


20 


9 


8 


53 


9 


671 


807 


164 


204 


1881 


9 


9 


16 


1 


9 


8 


54 


10 


031 


722 


790 


257 


1882 


9 


9 


1 


41 


9 


8 


55 


12 


391 


637 


415 


311 


1883 


9 


8 


47 


22 


9 


8 


56 


13 


751 


552 


040 


364 


1884 B. 


9 


9 


32 


10 


9 


8 


57 


15 


145 


469 


666 


418 


1885 


9 


9 


17 


51 


9 


8 


58 


16 


505 


385 


292 


471 


1886 


9 


9 


3 


32 


9 


8 


59 


17 


865 


300 


918 


525 


1887 


9 


8 


49 


12 


9 


8 





19 


225 


216 


544 


579 


1888 B. 


9 


9 


34 


1 


9 


8 


1 


20 


619 


133 


169 


6 



21 



10 



TABLE V. 

sun's motions for months. 



Months. 


Longitude. 


Per. 


I. 


II. 


III. 


N. 


T "1 Com. . . . 
Jan -JBis 

Feb.]S? m ' •'• 
J Bis 


s. ° ' " 


11 29 52 

1 33 18 

29 34 10 

1 28 9 11 




5 
5 
10 




966 

47 

13 

993 




997 

78 

75 

148 




998 

53 

51 

01 




4 
4 
9 


April 


2 28 42 30 

3 28 16 40 

4 28 49 58 

5 28 24 8 

6 28 57 26 


15 
20 
26 
31 
36 


42 

59 

110 

129 

182 


226 
301 
379 
454 
531 


154 
206 
259 
310 
363 


13 
18 
22 
27 
31 


May 


J une 


Julv 


August 


7 29 30 44 

8 29 4 54 

9 29 38 12 
10 29 12 22 


41 
46 
52 
57 


233 
250 
300 
313 


609 

684 
762 

837 


416 
468 
521 
572 


36 
40 
45 
49 





TABLE VI 



SUN S HOURLY MOTION. 
Argument. — Sun's Mean Anomaly. 





0s 


Is 


lis 


Ills 


IVs 


V 




o 

10 
20 
30 


2 33 
2 33 
2 33 
2 32 


2 32 
2 32 
2 31 

2 30 


/ // 

2 30 
2 29 
2 29 
2 28 


/ // 

2 28 
2 27 
2 26 
2 25 


/ // 

2 25 
2 25 
2 24 
2 24 


2 24 
2 23 
2 23 
2 23 


o 
30 
20 
10 





XIs 


Xs 


IXs 


VIIIs 


VIIs 


Vis 



SUN S SEMLDIAMETER. 
Argument. — Sun's Mean Anomaly. 



o 


10 
20 
30 


0s 


Is 


lis 


Ills 


IVs 


Vs 




16 18 
16 18 
16 17 
16 15 


/ // 

16 15 
16 14 
16 12 
16 9 


16 9 
16 7 
16 4 
16 1 


/ // 

16 1 
15 58 
15 56 
15 53 


/ // 

15 53 
15 51 
15 49 

15 48 


/ // 

15 48 
15 46 
15 46 
15 45 


o 
30 
20 
10 



■ 


XIs 


Xs 


IXs 


VIIIs 


VIIs 


Vis 





TABLE VII. 
sun's motions foe days and hours. 



11 



! Days. 

i 
i 


Logitude. 


Per. 


I. 


II. 


III. 


N. 




Hours. 


Long. 


I. j 


! 
i 


O / " 





















1 


2 28 


i ; 


2 


59 8 





34 


3 


2 







2 


4 56 


3 


3 


1 58 17 





68 


5 


3 







3 


7 23 


4 


4 


2 57 25 





101 


8 


5 







4 


9 51 


6 


5 


3 56 33 


1 


135 


10 


7 


1 




5 


]2 19 


7 ! 


6 


4 55 42 




169 


13 


9 


1 




6 


14 fa 


s ; 


7 


5 54 50 




203 


15 


10 


1 




7 


17 15 


10 


8 


6 53 58 




236 


18 


12 


1 




8 


19 43 


11 ! 


9 


7 53 7 




270 


20 


14 


1 




9 


22 11 


13 


10 


8 52 15 




304 


23 


15 


1 




10 


24 38 


14 i 


11 


9 51 23 


2 


338 


25 


17 


1 




11 


27 6 


1 
16 , 


12 


10 50 32 


2 


371 


28 


19 


2 




12 


29 34 


17 


13 


11 49 40 


2 


405 


30 


21 


2 




13 


32 2 


18 


14 


12 48 48 


2 


439 


33 


22 


2 




14 


34 30 


20 


15 


13 47 57 


2 


473 


35 


24 


2 




15 


36 58 


21 


16 


14 47 5 


o 


506 


38 


26 


2 




16 


39 26 


23 


17 


15 46 13 


3 


540 


40 


27 


2 




17 


41 53 


24 


18 


16 45 22 


3 


574 


43 


29 


2 




18 


44 21 


25 


19 


17 44 30 


3 


608 


45 


31 


3 




19 


46 49 


27 ; 


20 


18 43 38 


3 


641 


48 


33 


3 




20 


49 17 


28 


21 


19 42 47 


3 


675 


50 


34 


3 




21 


51 45 


30 : 


22 


20 41 55 


4 


709 


53 


36 


3 




22 


54 13 


31 


23 


21 41 3 


4 


743 


55 


38 


3 




23 


56 40 


32 


24 


22 40 12 


4 


777 


58 


39 


3 




24 


59 8 


34 


25 


23 39 20 


4 


810 


60 


41 


4 








f 

i 


26 


24 38 28 


4 


844 


63 


43 


4 










27 


25 37 37 


4 


-878 


65 


45 


4 










28 


26 36 45 


5 


912 


68 


46 


4 










29 


27 35 53 


5 


945 


70 


48 


4 








j 

! 


30 


28 35 2 


5 


979 


73 


50 


4 








i 


31 


29 34 10 


5 


13 


75 


51 


4 








! 



SUN S MOTIONS FOR MINUTES. 



Min. 


Longitude. 


Min. 


Longitude. 


1 


2 


30 


1 16 


5 


12 


35 


1 26 


10 


25 


40 


1 39 


15 


37 


45 


1 51 


20 


49 


50 


2 3 


25 


1 2 


55 


2 16 


30 


1 14 


j 60 


2 28 



2a 



ID 



TABLE VIII. 



EQUATIONS OF THE SUN S CENTER. 
Argument. — Sun's Mean Anomaly. 





0s 


Is 


lis 


Ills 


IVs 


n 
Vs 


o 


o 


/ 


n 


o 


i 


a 


o 


i 


a 


o 


i 


a 


o 


/ 


ii 


o 


/ // 





1 


59 


30 


2 


58 


15 


3 


40 


27 


3 


54 


50 


3 


38 


21 


2 


56 9 


1 


2 


1 


33 


3 








3 


41 


25 


3 


54 


47 


3 


37 


18 


2 


54 25 


2 


2* 


3 


37 


3 


1 


44 


3 


42 


21 


3 


54 


41 


3 


36 


14 


2 


52 40 


3 


2 


5 


40 


3 


3 


27 


3 


43 


15 


3 


54 


33 


3 


35 


8 


2 


50 54 


4 


2 


7 


43 


3 


5 


9 


3 


44 


8 


3 


54 


23 


3 


34 


1 


2 


49 8 


5 


2 


9 


46 


3 


6 


49 


3 


44 


58 


3 


54 


11 


3 


32 


51 


2 


47 20 


6 


2 


11 


49 


3 


8 


28 


3 


45 


47 


3 


53 


57 


3 


31 


41 


2 


45 32 


7 


2 


13 


51 


3 


10 


6 


3 


46 


33 


3 


53 


41 


3 


30 


28 


2 


43 43 


8 


2 


15 


54 


3 


11 


43 


3 


47 


17 


3 


53 


23 


3 


29 


14 


2 


41 53 


9 


2 


17 


56 


3 


13 


18 


3 


48 





3 


53 


3 


3 


27 


58 


2 


40 3 


10 


2 


19 


57 


3 


14 


51 


3 


48 


40 


3 


52 


40 


3 


26 


41 


2 


38 11 


11 


2 


21 


58 


3 


16 


24 


3 


49 


18 


3 


52 


16 


3 


25 


22 


2 


36 19 


12 


2 


23 


59 


3 


17 


54 


3 


49 


55 


3 


51 


50 


3 


24 


2 


2 


34 27 


13 


2 


25 


59 


2 


19 


24 


3 


50 


29 


3 


51 


21 


3 


22 


40 


2 


32 34 


14 


2 


27 


59 


3 


20 


51 


3 


51 


1 


3 


50 


51 


3 


21 


17 


2 


30 40 


15 


2 


29 


58 


3 


22 


18 


3 


51 


31 


3 


50 


18 


3 


19 


52 


2 


28 46 


16 


2 


31 


57 


3 


23 


42 


3 


51 


59 


3 


49 


44 


3 


18 


26 


2 


26 52 


17 


2 


33 


55 


3 


25 


5 


3 


52 


25 


3 


49 


7 


3 


16 


58 


2 


24 56 


18 


2 


35 


52 


3 


26 


26 


3 


52 


49 


3 


48 


29 


3 


15 


30 


2 


23 


19 


2 


37 


49 


3 


27 


46 


3 


53 


10 


3 


47 


49 


3 


14 





2 


21 4 


30 


2 


39 


45 


3 


29 


4 


3 


53 


30 


3 


47 


7 


3 


12 


28 


2 


19 8 


21 


2 


41 


40 


3 


30 


24 


3 


53 


47 


3 


46 


22 


3 


10 


55 


2 


17 11 


22 


2 


43 


34 


3 


31 


35 


3 


54 


3 


3 


45 


36 


3 


9 


22 


2 


15 14 


23 


2 


45 


28 


3 


32 


48 


3 


54 


16 


3 


44 


48 


3 


7 


46 


2 


13 16 


24 


2 


47 


20 


3 


33 


59 


3 


54 


27 


3 


43 


58 


3 


6 


10 


2 


11 19 


25 


2 


49 


12 


3 


35 


8 


3 


54 


36 


3 


43 


7 


3 


4 


33 


2 


9 21 


26 


2 


51 


2 


3 


36 


16 


3 


54 


43 


3 


42 


13 


3 


2 


54 


2 


7 23 


27 


2 


52 


52 


3 


37 


21 


3 


54 


48 


3 


41 


18 


3 


1 


14 


2 


5 25 


28 


2 


54 


41 


3 


38 


25 


3 


54 


51 


3 


40 


21 


2 


59 


33 


2 


3 27 


29 


2 


56 


28 


3 


39 


27 


3 


54 


52 


3 


39 


22 


2 


57 


52 


2 


1 28 


30 


2 


58 


15 


3 


40 


27 


3 


54 


50 


3 


38 


21 


2 


56 


9 


1 


59 30 

] 



TABLE VIII. 



13 



EQUATIONS OF THE SXJN's CENTER. 
Argument. — Sun's Mean Anomaly. 





Vis 


VIIs 


VIIIs 


IXs 


Xs 


XIs 


o 


o 


i 


n 


o 


1 


// 


o 


/ 


a 


o 


1 


a 


O i 


i' 


o 


i a 





1 


59 


30 


1 


2 


51 





20 


39 





4 


10 


18 


33 


1 


45 


1 


1 


57 


32 


1 


1 


8 





19 


38 





4 


8 


19 


33 


1 


2 32 


2 


1 


55 


33 





59 


27 





18 


39 





4 


9 


20 


35 


1 


4 19 


3 


1 


53 


35 





57 


46 





17 


42 





4 


12 


21 


39 


1 


6 8 


4 


1 


51 


37 





56 


6 





16 


47 





4 


17 


22 


44 


1 


7 58 


5 


1 


49 


39 





54 


27 





15 


53 





4 


24 


23 


52 


1 


9 48 


6 


1 


47 


41 





52 


47 





15 


2 





4 


33 


25 


1 


1 


11 40 


7 


1 


45 


44 





51 


14 





14 


12 





4 


44 


26 


12 


1 


13 32 


8 


1 


43 


46 





49 


38 





13 


24 





4 


57 


27 


25 


1 


15 26 


9 


1 


41 


49 





48 


5 





12 


38 





5 


13 


28 


40 


1 


17 20 


10 


1 


39 


52 





46 


32 





11 


53 





5 


30 


29 


56 


1 


19 15 


11 


1 


37 


56 





45 








11 


11 





5 


50 


31 


14 


1 


21 11 


12 


1 


36 








43 


30 





10 


31 





6 


11 


32 


34 


1 


23 8 


13 


1 


34 


4 





42 


1 





9 


53 





6 


35 


33 


55 


1 


25 5 


14 


1 


32 


9 





40 


34 





9 


16 





7 


1 


35 


18 


1 


27 3 


15 


1 


30 


14 





39 


8 





8 


42 





7 


29 


36 


42 


1 


29 2 


16 


1 


28 


20 





37 


43 





8 


9 





7 


59 


38 


9 


1 


31 1 


17 


1 


26 


26 





36 


20 





7 


39 





8 


31 


39 


36 


1 


33 1 


18 


1 


24 


33 





34 


58 





7 


10 





9 


5 


41 


9 


1 


35 1 


19 


1 


22 


41 





33 


38 





6 


44 





9 


42 


42 


36 


1 


37 1 


20 


1 


20 


49 





32 


19 





6 


20 





10 


20 


44 


9 


1 


39 3 


21 


1 


18 


57 





31 


2 





5 


57 





11 





45 


42 


1 


41 4 


22 


1 


17 


7 





29 


46 





5 


37 





11 


43 


47 


17 


1 


43 6 


23 


1 


15 


17 





28 


32 





5 


19 





12 


27 


48 


54 


1 


45 9 


24 


1 


13 


28 





27 


19 





5 


3 





13 


13 


50 


32 


1 


47 11 


25 


1 


11 


40 





26 


9 





4 


49 





14 


2 


52 


11 


1 


49 14 


26 


1 


9 


52 





24 


59 





4 


37 





14 


52 


53 


51 


1 


51 17 


27 


1 


8 


6 





23 


52 





4 


27 





15 


45 


55 


33 


1 


53 20 


28 


1 


6 


20 





22 


46 





4 


19 





16 


39 


57 


16 


1 


55 23 


29 


1 


4 


35 





21 


41 





4 


13 





17 


35 


59 





1 


57 27 


30 


1 


2 


51 





20 


39 





4 


10 





18 


33 


1 


45 


1 


59 30 | 



14 



TABLE IX. 

SMALL EQUATIONS OF THE SUN'S LONGITUDE. 



Arg. 


I 


II. 


III. 


Arg. 


I. 


II. 


III. 




a 


a 


II 




a 


a 


// 





10 


10 


10 


500 


10 


10 


10 


10 


10 


11 


9 


510 


10 


10 


9 


20 


11 


11 


9 


520 


9 


10 


8 


30 


11 


12 


8 


530 


9 


10 


7 


40 


11 


13 


8 


540 


9 


10 


7 


40 


12 


14 


7 


550 


8 


10 


6 


60 


12 


14 


7 


560 


8 


9 


5 


70 


12 


15 


7 


570 


8 


9 


4 


80 


13 


15 


7 


580 


7 


9 


3 


90 


13 


16 


7 


590 


7 


9 


3 


100 


13 


16 


7 


600 


7 


9 


2 


110 


14 


17 


7 


610 


6 


8 


1 


120 


14 


17 


7 


620 


6 


8 


1 


130 


14 


18 


8 


630 


6 


8 


1 


140 


15 


18 


8 


640 


5 


7 





150 


15 


18 


9 


650 


5 


7 





160 


15 


18 


9 


660 


5 


6 





170 


15 


18 


10 


670 


5 


6 


1 


180 


15 


18 


10 


680 


5 


6 


1 


190 


16 


18 


11 


690 


4 


5 


2 


200 


16 


18 


11 


700 


4 


5 


2 


210 


16 


18 


12 


710 


4 


4 


3 


220 


16 


18 


12 


720 


4 


4 


3 


230 


16 


18 


13 


730 


4 


4 


4 


240 


16 


17 


14 


740 


4 


3 


5 


250 


16 


17 


14 


750 


4 


3 


6 


260 


16 


17 


15 


760 


4 


3 


6 


270 


16 


16 


16 


770 


4 


2 


7 


480 


16 


16 


17 


780 


4 


2 


8 


290 


16 


16 


17 


790 


4 


2 


8 


300 


16 


15 


18 


800 


4 


2 


9 


310 


16 


15 


18 


810 


4 


2 


9 


320 


15 


14 


19 


820 


5 


2 


10 


330 


15 


14 


19 


830 


5 


2 


10 


340 


15 


14 


20 


840 


5 


2 


11 


350 


15 


13 


20 


850 


5 


2 


11 


360 


15 


13 


20 


860 


5 


2 


12 


370 


14 


12 


19 


870 


6 


2 


12 


380 


14 


12 


19 


880 


6 


3 


13 


390 


14 


12 


19 


890 


6 


3 


13 


400 


13 


11 


18 


900 


7 


4 


13 


410 


13 


11 


17 


910 


7 


4 


13 


420 


13 


11 


17 


920 


7 


5 


13 


430 


12 


11 


16 


930 


8 


5 


13 


440 


12 


11 


15 


940 


8 


6 


13 


450 


12 


10 


14 


950 


8 


6 


13 


460 


11 


10 


13 


960 


9 


7 


12 


470 


11 


10 


13 


970 


9 


8 


12 


480 


11 


10 


12 


980 


9 


9 


11 


490 


10 


10 


11 


990 


10 


9 


11 


500 


10 


10 


10 


1000 


10 


10 


10 



TABLE X. 



15 



NUTATIONS. 
Argument. — Supplement of the Node, or N. 



N. 


Long. 


R. Asc. 


Obliq. 

i 


N. 


Long. 


R. Asc. 


Obliq. 





+ ° 


4- ° 


I 

4- io 


500 


— 


— 


II 

— 10 


20 


2 


2 


10 


520 


2 


2 


9 


40 


4 


4 


9 


540 


4 


4 


9 


60 


7 


6 


9 


560 


7 


6 


9 


80 


9 


8 


8 


580 


9 


8 


8 


100 


+ 11 


4- 10 


4- 8 


600 


— 11 


— 10 


— 8 


120 


12 


11 


T 7 


620 


12 


11 


t 


140 


14 


13 


6 


640 


14 


13 


6 


160 


15 


14 


5 


660 


15 


14 


5 


180 


16 


15 


4 


680 


16 


15 


4 


200 


-f-17 


4- 16 


+ 3 
1 2 


700 


— 17 


— 16 


— 3 


220 


18 


T 16 


720 


18 


16 


2 


240 


18 


16 


1 


740 


18 


16 


1 


260 


18 


16 


_ 1 


760 


18 


16 


+ 1 


280 


18 


16 


2 


780 


18 


16 


2 


300 


+ 17 


4- 16 


_ 3 


800 


— 17 


— 16 


+ 3 


320 


16 


^ 15 


4 


820 


16 


15 


4 


340 


15 


14 


5 


840 


15 


14 


5 


360 


14 


13 


6 


860 


14 


13 


6 


380 


12 


11 


7 


880 


12 


11 


7 


400 


+ 11 


4- 10 


— 8 


900 


— 11 


— 10 


+ 8 


420 


9 


T 8 


8 


920 


9 


8 


8 


440 


7 


6 


9 


940 


7 


6 


9 


460 


4 


4 


9 


960 


4 


4 


9 


480 


2 


2 


10 


980 


2 


2 


10 


500 


+ o 


+ o 


— 10 


1000 


— 


— 


4-10 



TABLE XI. 

Earth's Radius Vector. — Argument. Sun's Mean Anomaly. 



0o 


Os 


Is 


lis 


Ills 


IV s 


Vs 


30° 


0.98313 


0.98545 


0.99173 


1.00018 


1.00850 


1.01450 


2 


0.98314 


0.98576 


0.99225 


1.00077 


1.00899 


1.01477 


28 


4 


0.98317 


0.98608 


0.99278 


1.00135 


1.00947 


1.01503 


26 


6 


0.98322 


0.98643 


0.99331 


1.10193 


1.00994 


1.01527 


24 


8 


0.98330 


0.98679 


0.99386 


1.00251 


1.01040 


1.01549 


22 


10 


0.98339 


0.98717 


0.99441 


1.00308 


1.01084 


1.01569 


20 


12 


0.98350 


0.98756 


0.99497 


1.00366 


1.01128 


1.01588 


18 


14 


0.98364 


0.98797 


0.99554 


1.00422 


1.01170 


1.01604 


16 


16 


0.98380 


0.98840 


0.99611 


1.00478 


1.01210 


1.01619 


14 


18 


0.98397 


0.98883 


0.99668 


1.00534 


1.01249 


1.01632 


12 


20 


0.98417 


0.98929 


0.99726 


1.00588 


1.01286 


1.01643 


10 


22 


0.9843-9 


0.98975 


0.99784 


1.00642 


1.01322 


1.01652 


8 


24 


0.98462 


0.99023 


0.99843 


1.00695 


1.01357 


1.01659 


6 


26 


0.98486 


0.99072 


0.99901 


1.00748 


1.01389 


1.01663 


4 


28 


0.98515 


0.99122 


0.99960 


1.00799 


1.01420 


1.01666 


2 


30 


0.98545 


0.99173 


1.00018 


1.00850 


1.01450 


1.01667 







XIs 


Xs 


lXs 


vras 


VIIs 


Vis 



2a* 



16 



TABLE XL 



MEAN NEW MOONS AND ARGUMENTS IN JANUARY. 



Mean New 
Moon in 
January. 



A. D. 

1836 B. 
1837 

1838 
1839 
1840 B. 

1841 

1842 
1843 

1844 B. 
1845 

1846 

1847 
1848 B. 
1849 
1850 

1851 
1852 B. 
1853 
1854 
1855 

1856 B. 

1857 
1858 
1859 
1860 B. 

1861 

1862 
1863 
1864 B. 
1865 

1866 

1867 
1868 B. 
1869 
1870 

1871 
1872 B. 
1873 
1874 
1875 

1876 B. 

1877 
1878 
1879 
1880 B. 



D. H. M. 

17 10 32 

5 19 20 

24 16 53 



14 
3 



1 42 
10 30 



21 8 3 
10 16 51 
29 14 24 
18 23 13 

7 8 1 

26 5 34 
15 14 22 

4 23 11 

22 20 43 

12 5 32 

1 14 21 
20 11 53 

8 20 42 

27 18 14 

17 3 3 

6 11 51 

24 9 24 

13 18 13 
3 3 1 

22 34 

10 9 22 
29 6 55 

18 15 44 
8 32 

25 22 5 



15 

4 

23 

11 

1 



6 53 
15 42 
13 14 
22 3 

6 51 



20 4 24 
8 13 13 

27 10 46 
17 19 35 

7 4 24 

26 1 57 

14 10 49 

3 18 38 

22 6 11 

11 15 



0469 
0171 
0681 
0383 

0085 

0595 
0297 
0807 
0509 
0211 

0721 
0423 
0125 
0635 
0337 

0038 
0549 
0251 
0761 
0463 

0164 
0675 
0376 

0078 
0588 

0290 
0800 
0504 
0204 
0714 

0416 
0118 
0628 
0330 
0032 

0542 
0244 
0754 

0456 

0158 

0668 
0370 
0072 

0582 
0284 



II. 



1246 
9852 
9175 
7780 
6386 

5709 
4314 
3637 
2243 
0848 

0171 

8777 
7382 
6705 
5311 

3916 
3239 
1845 
1168 
9773 

8379 
7702 
6307 
4913 
4236 

2840 
2L63 
0769 
9374 
8698 

7303 
5909 
5231 
3837 
2442 

1765 
0371 
9694 
8300 
6906 

6229 

4835 
3441 
2764 
1370 



in. 
17 


IV. 


N. 


08 


669 


00 


97 


692 


99 


85 


799 


82 


74 


822 


65 


63 


844 


63 


51 


951 


46 


40 


974 


44 


28 


081 


28 


17 


104 


11 


06 


126 


09 


94 


234 


92 


84 


256 


75 


73 


278 


73 


61 


386 


56 


50 


408 


40 


39 


431 


38 


27 


538 


21 


16 


560 


19 


04 


668 


02 


93 


690 


85 


82 


713 


84 


70 


820 


1 67 


59 


843 


50 


48 


865 


48 


36 


972 


31 


25 


995 


30 


14 


102 


13 


03 


125 


96 


92 


147 


94 


80 


256 


77 


69 


277 


60 


58 


299 


59 


46 


407 


42 


35 


429 


25 


24 


451 


23 


12 


559 


05 


01 


581 


03 


89 


689 


86 


78 


711 


69 


67 


733 


67 


55 


841 


50 


44 


863 


33 


23 


885 


31 


21 


993 


J 14 


10 


015 



TABLE XII 



17 



MEAN LUNATIONS AND CHANGES OP THE ARGUMENTS. 



Num. 


Lunations. 


I. 


II. 


III. 


IV. 


N. 




d. 


h. 


m. 












y, 


14 


18 


22 


404 


5359 


58 


50 


43 


i 


29 


12 


44 


808 


717 


15 


99 


85 


2 


59 


1 


28 


1617 


1434 


31 


98 


170 


3 


88 


14 


12 


2425 


2151 


46 


97 


256 


4 


118 


2 


56 


3234 


2869 


61 


96 


341 


5 


147 


15 


40 


4042 


3586 


76 


95 


425 


6 


177 


4 


24 


4851 


4303 


92 


95 


511 


7 


206 


17 


8 


5659 


5020 


7 


94 


596 


8 


236 


5 


52 


6468 


5737 


22 


93 


682 


9 


265 


18 


36 


7276 


5454 


37 


92 


767 


10 


295 


7 


20 


8085 


7117 


53 


91 


852 


11 


324 


20 


5 


8893 


7889 


68 


90 


937 


12 


354 


8 


49 


9702 


8606 


83 


89 


22 


13 


383 


21 


33 


510 


9323 


93 


88 


108 



TABLE XIII. 



TABLE XIV. 



NUMBER OF DATS FROM THE 
COMMENCEMENT OF THE YEAR 
TO THE FIRST OF EACH MONTH. 



Months. 


Com. 


Bis. 


January. . . 
February. . 
March 

May 

July 

August.. . . 
September. 
October . . . 
November. 
December . 


Days. 



31 

59 

90 

120 

151 

181 

212 

243 

273 

304 

334 


Days. 


31 

60 

91 
121 
152 
182 
213 
244 
274 
305 
335 j 



Arg. 


( 


1 


(§ 


d 


Arg. 


II. 


H. Par. 


S.D. 


H. 


Mo. 


II. 





60 


29 


16 


29 


/ 
36 


// 

48 


10000 


250 


60 


26 


16 


26 


36 


44 


9750 ! 


500 


60 


17 


16 


25 


36 


19 


9500 i 


750 


60 





16 


21 


36 


8 


9250 i 


1000 


59 


47 


16 


17 


35 


48 


9000 i 


1250 


59 


24 


16 


11 


35 


28 


8750 ! 


1500 


58 


56 


16 


3 


34 


57 


8500 I 


1750 


58 


30 


15 


56 


34 


34 


8250 1 


2000 


58 


7 


15 


50 


33 


58 


8000 j 


2250 


57 


37 


15 


42 


33 


32 


7750 I 


2500 


57 


1 


15 


31 


32 


42 


7500 


2750 


56 


32 


15 


23 


32 


9 


7250 


3000 


56 


2 


15 


16 


31 


36 


7000 ! 


3250 


55 


40 


15 


10 


31 


13 


6750 


3500 


55 


22 


15 


7 


30 


52 


6500 


3750 


55 


12 


15 


3 


30 


29 


6250 


4000 


54 


51 


14 


56 


30 


7 


6000 j 


4250 


54 


39 


14 


54 


29 


55 


5750 


4500 


54 


26 


14 


50 


29 


43 


5500 , 


4750 


54 


18 


14 


48 


29 


37 


5250 


5000 


54 


13 


14 


45 


29 


35 


5000 | 



18 



TABLE XV. 

EQUATIONS FOE, NEW AND FULL MOON. 



Arg. 


I. 


II. 


Arg. 


I. 


II. 
h. m. 


Arg. 


III. 
m. 


IV. 


Arg. 




h. m. | 


h. m. 




h. m. 




m. 







4 20 


10 10 


5000 


4 20 


10 10 


25 


3 


31 


25 


100 


4 36 ! 


9 36 


5100 


4 5 


10 50 


26 


3 


31 


24 


200 


4 52 


9 2 


5200 


3 49 


11 30 


27 


3 


30 


23 


> 300 


5 8 


8 28 


5300 


3 34 


12 9 


28 


3 


30 


22 


400 


5 24 


7 55 


5400 


3 19 


12 48 


29 


3 


30 


21 


500 


5 40 


7 22 


5500 


3 4 


13 26 


30 


3 


30 


20 


600 


5 55 


6 49 


5600 


2 49 


14 3 


31 


3 


30 


19 


700 


6 10 


6 17 


5700 


2 35 


14 39 


32 


4 


20 


18 


800 


6 24 


5 46 


5800 


2 21 


15 13 


33 


4 


29 


17 


900 


6 38 


5 15 


5900 


2 8 


15 46 


34 


4 


29 


16 


1000 


6 51 


4 46 


6000 


1 55 


16 18 


35 


4 


29 


15 


1100 


7 4 


4 17 


6100 


1 42 


16 48 


36 


5 


28 


14 


1200 


7 15 


3 50 


6200 


1 31 


17 16 


37 


5 


28 


13 


1300 


7 27 


3 24 


6300 


1 19 


17 42 


38 


5 


27 


12 


1400 


7 37 


2 59 


6400 


1 9 


18 6 


39 


5 


27 


11 


1500 


7 47 


2 35 


6500 


59 


18 28 


40 


6 


26 


10 


1600 


7 55 


2 14 


6600 


50 


18 48 


41 


6 


26 


9 


! 1700 


8 3 


1 53 


6700 


42 


19 6 


42 


7 


25 


8 


| 1800 


8 10 


1 35 


6800 


34 


19 21 


43 


7 


25 


7 


; 1900 


8 16 


1 38 


6900 


28 


19 33 


44 


7 


24 


6 


i 2000 


8 21 


1 3 


7000 


22 


19 44 


45 


8 


23 


5 


i 2100 


8 25 


51 


7100 


17 


19 52 


46 


8 


23 


4 


' 2200 


8 29 


40 


7200 


14 


19 57 


47 


9 


22 


3 


! 2300 


8 31 


32 


7300 


11 


20 


48 


9 


21 


2 


i 2400 


8 32 


25 


7400 


9 


20 1 


49 


10 


21 


1 


i 2500 


8 32 


21 


7500 


8 


19 59 


50 


10 


20 





! 2600 


8 31 


19 


7600 


.0 8 


19 55 


51 


10 


19 


99 


! 2700 


8 29 


20 


7700 


9 


19 48 


52 


11 


19 


98 


! 2300 


8 26 


23 


7800 


11 


19 40 


53 


11 


18 


97 


' 2900 


8 23 


28 


7900 


15 


19 29 


54 


12 


17 


96 


: sooo 


8 18 


36 


8000 


19 


19 17 


55 


12 


17 


95 


3100 


8 12 


47 


8100 


24 


19 2 


56 


13 


16 


94 


3200 


8 6 


59 


8200 


30 


18 45 


57 


13 


15 


93 


3300 


7 58 


1 14 


8300 


37 


18 27 


58 


13 


15 


92 


3400 


7 50 


1 32 


8400 


45 


18 6 


59 


14 


14 


91 


3500 


7 41 


1 52 


8500 


53 


17 45 


60 


14 


14 


90 


3600 


7 31 


2 14 


8600 


1 3 


17 21 


61 


15 


13 


89 


3700 


7 21 


2 38 


8700 


1 13 


16 56 


62 


15 


13 


88 


3800 


7 9 


3 4 


8800 


1 25 


16 30 


63 


15 


12 


87 


3900 


6 58 


3 32 


8900 


1 36 


16 3 


64 


15 


12 


86 


| 4000 


6 45 


4 2 


9000 


1 49 


15 34 


65 


16 


11 


85 


j 4100 


6 32 


4 34 


9100 


2 2 


15 5 


66 


16 


11 


84 


! 4200 


6 19 


5 7 


9200 


2 16 


14 34 


67 


16 


11 


83 


1 4300 


6 5 


5 41 


9300 


2 30 


14 3 


68 


16 


10 


82 


4400 


5 51 


6 17 


9400 


2 45 


13 31 


69 


17 


10 


81 


4500 


5 36 


6 54 


9500 


3 


12 58 


170 


17 


10 


80 


4600 


5 21 


7 32 


9600 


3 16 


12 25 


71 


17 


10 


79 


4700 


5 6 


8 11 


9700 


3 32 


11 52 


72 


17 


10 


78 


4800 


4 51 


8 50 


9800 


3 48 


11 18 


73 


17 


10 


77 


4900 


4 35 


9 30 


9900 


4 4 


10 44 


74 


17 


9 


76 


5000 


4 20 


10 10 

i ! 


10000 


4 20 


10 10 


75 


17 


9 


75 

> 



TABLE E. 



19 



CD 

S3 



p 

£2 

p s 



=- CD 
CO "■* 

| o 

% ° 

C o 



CP3 
3 



S" tr 



p P 



pi 


re 


— 


P 




s 


p 








0". 


R 


o 


CD 


s 




** 


P 




^ 






TJ 


^. 



r pi 






COtOtOtOtOtOi-'i-'H-ii-'l-' 

©eoos^toocoas^tooooas^ioo 



J -U -]_ 

to tO Cn to CO I— '^(OOi^HUiW.. 
^0000H-'COCntO©CO©Cn>JSi.COpasH-S Q 

co^bs^©csi^i^io^a;'^Goao©io 



COCOCOCOCOCOCOCOCOCOtOtOtOI-'l-'l-'B' 

£^£vCnencncn4s».totO oiwmww _ 
iO©CO^^©^tOtO©enCnptOtO;-3. 

en^tocoioasJ-'aootnen © oo as £^ co 



+ 



l-'00000^ 4 l- i 



tOtOtOtOeoCOCOCo3 



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Cn cn oc 

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to tO 0^ i- 1 
p jt*. CD £>. 

bwbbo 



^ © 



tO CO Cn l- 1 JO 

p 00 <i to cn 

'&> © f- 1 en © 



to rfa. 
tn to 

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C50JmmUiOi*kit».ifckCOWWWlO 






Or 

P° 'r' ^ 

tO Mi t-J 



OS CO O JO OS 



rf^ to i— ' en 
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tototoco^^enenenencnasasas 



lO Oi ts 



to "<J t— ' O 

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o to as oo to oo p 

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1 + 



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to £*. 

J . 10 

^cnbocobbbbw 



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< 



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to to cn to ^i— irfx to en i— ■ eo en h-" to r/) 
ppp^oototop^toppopopopp- 
b<it.Mcck)I- , bb^.<!bMcoool|i 



<i 



coH^^enoicncnasascsascsascstncnif 1 

to to i— 'rf^en i-" i— 'i— ■ i— ' rfuto r/5 
©©cs©j-'©toco>-'cnp^©©aoco. 
^^issb^o^lobo^eotntoolf^^to 



HfOW^.tiS)<!^fQOtOOH-'H^WW 



to to 

to CD 

as io 



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to to 



en rfx to 
to -a oo -a 



to en to 

£». to i— ' en 



too © © £t I- 4 o to i— ■ 



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to 

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to to 



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p^Jpp^tOptOF- 1 
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-aacoc©©©©!— 'i— ' to to to co to to ^ 



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cn jo en L - ' 
to to to © 



torf*. 

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co ^ en 



pep? 

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> 

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ffi 


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g 


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3 


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1 


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3 


ffl 


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tr 1 


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a 2. 


> 


Ou 


HI 


CD 


»— i 




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o 




^ 



SI 
* 



20 








TABLE XVI. 












moon's epochs. 








1 Years. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


1846 


0013 


2475 


3275 


1688 


0773 


4880 


3179 


0800 


9542 


1847 


0006 


9683 


2941 


6432 


3245 


0678 


4239 


3257 


8406 


1848 B. 


O026 


7542 


3646 


1463 


6052 


6847 


5358 


6106 


7295 


1841 


0019 


4750 


3312 


6207 


8524 


2644 


6418 


8563 


6158 


1850 


0012 


1958 


2978 


0951 


0995 


8442 


7479 


1020 


5022 


1851 


0005 


9167 


2644 


5695 


3467 


4239 


8539 


3477 


3885 


1852 B. 


0025 


7025 


3350 


0726 


6274 


0408 


9658 


6326 


2774 


1853 


0018 


4233 


3016 


5469 


8746 


6206 


0718 


8782 


1637 


1854 


0011 


1442 


2681 


0213 


1217 


2003 


1778 


1240 


0501 


1855 


0004 


8650 


2347 


4957 


3689 


7801 


2839 


3697 


9365 


1856 B. 


0024 


6509 


3053 


9988 


6496 


3970 


3957 


6446 


8254 


1857 


0017 


3717 


2719 


4732 


8968 


9767 


5018 


9002 


7117 


!l858 


0010 


0925 


2385 


9476 


1439 


5565 


6078 


1460 


5981 


1859 


0003 


8134 


2051 


4220 


3911 


1362 


7139 


3917 


4845 


I860 B, 


0023 


5992 


2756 


9551 


6718 


7531 


8257 


6765 


3734 


11861 


0016 


3200 


2423 


3995 


9190 


3329 


9317 


9222 


2597 


:1862 


0009 


0409 


2088 


8739 


1661 


9126 


0378 


1679 


1461 


1863 


0002 


7617 


1754 


3483 


4133 


4923 


1438 


4137 


0324 


1864 B. 


0022 


5476 


2460 


8514 


6941 


1093 


2557 


6984 


9212 


'1865 


0015 


2684 


2126 


3257 


9412 


6890 


3617 


9442 


8076 


1866 


0008 


9893 


1792 


8001 


1883 


2687 


4678 


1899 


6940 


1867 


0001 


7101 


1457 


2745 


4355 


8485 


5738 


4357 


5804 


1868 B. 


0021 


4959 


2163 


7776 


7163 


4654 


6857 


7204 


4692 


1869 


0C14 


2168 


1829 


2520 


9634 


0452 


7917 


9662 


3556 


1870 


0007 


9376 


1495 


7264 


2105 


. 6249 


8978 


2119 


2420 


1871 


0000 


6584 


1161 


2008 


4576 


2046 


0039 


4576 


1284 


!l872 B. 


0020 


4432 


1867 


7039 


7383 


8215 


1158 


7423 


0172 


4873 


0013 


1640 


1533 


178-3 


9854 


4012 


2239 


9880 


9036 


1874 


0006 


8848 


1199 


6527 


2325 


9809 


3300 


2337 


7900 


1875 


9999 


6056 


0865 


1271 


4796 


5606 


4361 


4794 


6764 


11876 b. 


0019 


3914 


1571 


6292 


7603 


177o 


5480 


7641 


5652 


1877 


0012 


1122 


1247 


1036 


0074 


7572 


6541 


0098 


4516 


4878 


0005 


8330 


0913 


5780 


2545 


3369 


7602 


2555 


3380 


4879 


9998 


5538 


0579 


0524 


5016 


9166 


8663 


5012 


2244 


1880 B. 


0018 


3396 


1285 


5545 


7823 


5335 


9782 


7859 


1132 


1881 


0011 


0604 


0951 


0289 


0294 


1132 


0843 


0316 


9996 


1882 


0004 


7812 


0617 


5033 


2765 


6929 


1904 


2873 


8860 


1883 


9997 


5020 


0283 


9777 


5236 


2726 


2965 


5330 


7724 


1884 B. 


0017 


2878 


0989 


4798 


8043 


8895 


4084 


8177 


6612 


1885 


0010 


0086 


0655 


9542 


0514 


4692 


5145 


0634 


5476 


1886 


0003 


7294 


0321 


4286 


2985 


0489 


6206 


3091 


4340 | 


1887 


9996 


4502 


9987 


9030 


5456 


6286 


7267 


5548 


3204 


1888 B. 


0016 


2360 


0693 


4051 


8263 


2455 


8386 


8395 


2092 ! 


1889 | 


0009 


9568 


0359 


8795 


0734 1 


8252 


9447 


0852 1 


0956 | 


1890 j 


0002 ! 


6776 


0025 


3539 1 


3205 I 


4049 


0508 


3309 | 


9820 ! 



TABLE XVI. 



21 



moon's epochs. 



Years. 


10 


1 
11 1 12 

1 


13 


14 


15 


16 


17 


18 


19 


20 


1846 


203 


123 


250 


171 


419 


760 


126 


396 


167 


379 


204 


1847 


810 


484 


970 


644 


613 


901 


486 


749 


643 


433 


371 


1848 B. 


486 


876 


759 


151 


905 


072 


881 


143 


144 


487 


539 


1849 


093 


237 


479 


624 


099 


212 


241 


496 


619 


540 


705 


1850 


700 


597 


199 


097 


293 


352 


600 


848 


094 


594 


871 


1851 


306 


958 


918 


570 


487 


493 


960 


201 


569 


648 


038 


1852 B. 


983 


350 


707 


077 


780 


664 


355 


595 


070 


701 


206 


1853 


589 


711 


427 


550 


974 


804 


715 


948 


545 


755 


372 


1854 


196 


072 


147 


023 


168 


944 


074 


300 


020 


809 


539 


1855 


802 


432 


866 


496 


361 


085 


434 


653 


495 


863 


705 


1856 B. 


479 


824 


656 


003 


654 


256 


829 


047 


996 


916 


873 


1857 


086 


185 


375 


476 


848 


396 


189 


400 


471 


970 


039 


1858 


692 


546 095 


949 


042 


537 


548 


752 


947 


024 


206 


1859 


299 


907 ! 814 


422 


236 


677 


908 


105 


422 
923 


078 


372 ! 


1860 B. 


975 


298 


604 


929 


529 


848 


303 


499 


131 


540 


1861 


581 


659 


323 


402 


723 


988 


662 


852 


398 


185 706 


1862 


187 


020 


042 


875 


916 


]29 


021 


204 


873 


239 


873 


1863 


794 


381 


761 


348 


110 


269 


381 


557 


348 


292 


039 


1864 B. 


470 


773 | 551 


855 


403 


440 


777 


951 


849 


346 


207 


1865 


077 


134 271 


328 


597 


580 


136 


304 


324 


400 


373 


1866 


684 


494 j 990 


801 


791 


721 


495 


657 


799 


453 


540 


1867 


290 


855 \ 710 


274 


985 


861 


855 


009 


274 


507 


707 


1868 B. 


967 


247 ! 500 


781 


277 


032 


251 


404 


775 


561 


874 


1869 


573 


608 


219 


254 


471 


172 


610 


756 


251 


615 


040 


1870 


180 


968 


939 


737 


665 


313 


969 


109 


726 


668 


207 


1871 


787 


328 


659 


200 


859 


554 


328 


562 


201 


721 


374 


1872 B. 


464 


720 


549 


707 


151 


725 


724 


957 


702 


785 


531 


1873 


071 


080 i 269 


180 


345 


966 


083 


410 


177 


838 


698 


1874 


678 


440 


989 


653 


539 


295 


442 


863 


642 


891 


865 


1875 


285 


800 


709 


126 


733 


446 


801 


316 


117 


944 


032 


1876 B. 


962 


192 


599 


633 


025 


617 


197 


711 


618 


008 


199 


1877 


569 


552 


319 


106 


219 


858 


556 


164 


093 


061 


366 


1878 


176 


912 


039 


579 


413 


099 


915 


617 


568 


114 


533 


1879 


783 


272 


759 


052 


607 


340 


274 


070 


043 


167 


700 


1880 B. 


460 


664 


649 


559 


899 


511 


670 


465 


544 


231 


867 


1881 


067 


024 


369 


032 


093 


752 


029 


918 


019 


284 


034 


1882 


674 


384 


089 


505 


287 


993 


388 


371 


494 


337 


201 


1883 


281 


744 j 809 


978 


481 


234 


747 


824 


969 


390 


368 


1884 B. 


958 


136 699 


485 


773 


405 


143 


219 


470 


454 


535 


1885 


565 


496 j 419 


958 


967 


646 


502 


672 


945 


507 


702 


1886 


172 


856 139 


4S1 


161 


887 


861 


125 


420 


560 


869 


1887 


779 


216 


859 


904 


355 


128 


320 


578 


895 


613 


036 


1888 B. 


456 


608 


749 


411 


647 


299 


716 


973 


396 


677 


203 


1889 


063 


968 


469 


884 


841 540 


075 


426 


871 


730 


370 


1890 


670 


328 


189 


357 I 


035 1 781 


434 


879 


346 


783 


537 



22 



Months. 

T 1 Com. 
Jan. D . 

J Bis. 

Feb 1 Com * 
reD -J Bis. 

March 

April 

May 

June 

July 

August. . . . 

September . 
October. . . . 
November.. 
December. . 







TABLE XVIL 












moon's motions for months. 








1 


2 


3 


4 


5 


6 


7 


8 


9 


0000 


0000 


0000 


0000 


0000 


0000 


0000 


0000 


0000 


9973 


9350 


8960 


9713 


9664 


9628 


9942 


9610 


9976 


849 


146 


2246 


8896 


402 


1533 


1789 


2099 


753 


821 


9497 


1205 


8609 


66 


1161 


1731 


1709 


729 


1615 


8343 J 


1371 


6931 


9797 


1951 


3404 


3027 


1433 


2464 


8490 


3616 


5827 


199 


3484 


5193 


5126 


2186 


3285 


7986 


4822 


4436 


265 


4646 


6924 


6835 


2914 


4134 


8133 


7067 


3332 


666 


6179 


8713 


8934 


3667 


4955 


7629 


8273 


1942 


732 


7341 


444 


643 


4396 


5804 


7776 


518 


838 


1134 


8874 


2233 


2742 


5148 


6653 


7922 


2764 


9734 


1536 


408 


4021 


4842 


5901 


7474 


7419 


3969 


8343 


1602 


1569 


5752 


6550 


6630 


8323 


7565 


6215 


7239 


2004 


3102 


7541 


8649 


7382 


9144 


7062 


7420 


5848 


2070 


4264 


9272 


358 


8111 



TABLE XVIII. 

moon's motions for days. 



Days. 


1 


2 


3 


4 


5 


6 


7 


8 


1 9 ! 


1 


0000 


0000 


0000 


0000 


0000 


0000 


0000 


0000 


0000 


2 


27 


650 


1040 


287 


336 


372 


58 


390 


24 


3 


55 


1300 


2080 


574 


671 


744 


115 


781 


49 


4 


82 


3 950 


3121 


861 


1007 


1116 


173 


1171 


73 


5 


109 


2600 


4161 


1148 


1342 


1488 


231 


1561 


97 


6 


137 


3249 


5201 


1435 


1678 


1860 


289 


1952 


121 


7 


164 


3899 


6241 


1722 


2013 


2232 


346 


2342 


146 


8 


192 


4549 


7281 


2009 


2349 


2604 


404 


2732 


170 


9 


219 


5L99 


8321 


2296 


2684 


2976 


462 


3122 


194 


10 


246 


5849 


9362 


2583 


3020 


3348 


519 


3513 


219 


11 


274 


6499 


402 


2870 


3355 


3720 


577 


3903 


243 


12 


301 


7149 


1442 


3157 


3691 


4093 


635 


4293 


267 


13 


328 


7799 


2482 


3444 


4026 


4465 


692 


4684 


291 


1.4 


356 


8449 


3522 


3731 


4362 


4837 


750 


5074 


316 


15 


383 


9098 


4563 


4018 


4698 


5209 


808 


5464 


340 


16 


411 


9748 


5603 


4305 


5033 


5581 


866 


5854 


364 


17 


438 


398 


6643 


4592 


5369 


5953 


923 


6245 


389 


18 


465 


1048 


7683 


4878 


5704 


6325 


981 


6635 


413 


19 


493 


1698 


8723 


5165 


6040 


6697 


1039 


7025 


437 


20 


520 


2348 


9763 


5452 


6375 


7069 


1096 


7416 


461 


21 


548 


2998 


804 


5739 


6711 


7441 


1154 


7806 


486 


22 


575 


3648 


1844 


6026 


7046 


7813 


1212 


8196 


510 


23 


602 


4298 


2884 


6313 


7382 


8185 


1269 


8586 


534 


24 


630 


4947 


3924 


6600 


7717 


8557 


1327 


8977 


559 


25 


657 


5597 


4964 


6887 


8053 


8929 


1385 


9367 


583 


26 


684 


6247 


6005 


7174 


8389 


9301 


1443 


9757 


607 


27 


712 


6897 


7045 


7461 


8724 


9673 


1500 


148 


631 


28 


739 


7547 


8085 


7748 


9060 


45 


1558 


538 


656 


29 


767 


8197 


9125 


8035 


9395 


417 


1616 


928 


680 


30 


794 


8847 


165 


8322 


9731 


789 


1673 


1319 


704 ; 


31 


821 


9497 


1205 


8609 


66 


1161 


1731 


1709 


729 



TABLE XVII. 
moon's motions for months. 



23 



Months. 

T 1 Com 
Jan - J Brs. 

Feb.]£ om ' 
J Bis. 

March 



April. . , 
May . . , 
June . . 
July. . . 
August 



September . 
October.. . . 
November.. 
December. . 



10 



000 
930 
175 
105 
139 



314 
419 
593 

698 

873 

48 
152 
327 
432 



11 



000 
969 
965 
934 
836 

801 
735 
700 
634 
599 

563 
497 
462 
396 



12 



000 
930 
184 
114 
157 

342 
556 
640 
754 
938 

123 
237 
421 
535 



13 



000 

966 

59 

25 

16 

76 
101 
160 
185 
245 

304 
329 
388 
414 



14 



000 
901 
74 
975 
851 

925 
899 
973 
948 
22 

96 

71 

145 

120 



15 



000 
969 
946 
916 

801 



417 
333 
279 
194 



16 



000 
963 
135 
98 
159 



747 l 294 
663 I 392 
609 J 527 
525 I 625 
471 759 



894 
992 
127 
225 



17 



000 
958 
304 
262 

482 

786 
47 
351 
613 
917 

221 
483 

787 
49 



18 I 19 



000 
974 
805 
779 
532 

336 
115 

920 
699 
503 



20 



000 000 



000 
5 
5 
9 

13 
18 
22 
27 
31 



308 36 

87 40 

892 I 45 

670 I 49 



000 
14 
14 
27 

41 
55 
69 
83 
97 

111 
125 
139 
153 



TABLE XVIII. 

moon's motions for days. 



[Days. 

1 


10 


11 

000 


32 


13 


14 


15 


16 


17 


18 
000 


19 


20 


1 


000 


000 


000 


000 


000 


000 


000 


000 


000 


2 


70 


31 


70 


34 


99 


31 


37 


42 


26 








3 


140 


62 


141 


68 


198 


61 


73 


84 


52 





1 


4 


210 


93 


211 


103 


297 


92 


110 


126 


78 





1 


5 


281 


125 


282 


137 


397 


122 


146 


168 


104 


1 


2 


6 


351 


156 


352 


171 


496 


153 


183 


210 


130 


1 


2 


7 


421 


187 


423 


205 


595 


183 


220 


252 


156 


1 


3 


8 


491 


218 


493 


239 


694 


214 


256 


294 


182 


1 


3 


9 


561 


249 


564 


273 


793 


244 


293 


336 


208 


1 


4 


10 


6-31 


280 


634 


308 


892 


275 


329 


379 


234 


1 


4 


11 


702 


311 


705 


342 


992 


305 


366 


421 


260 


1 


5 


12 


772 


342 


775 


376 


91 


336 


403 


463 


236 


2 


5 


13 


842 


374 


845 


410 


190 


366 


439 


505 


312 


2 


5 


14 


912 


405 


916 


444 


239 


397 


476 


547 


337 


2 


6 


15 


982 


436 


986 


473 


388 


427 


512 


589 


363 


2 


6 


16 


52 


467 


57 


513 


487 


458 


549 


631 


389 


2 


7 I 


17 


122 


498 


127 


547 


587 


488 


586 


673 


415 


2 


7 ! 


18 


193 


529 


198 


581 


686 


519 


622 


715 


441 


2 


8 I 


19 


263 


560 


268 


615 


785 


549 


659 


757 


467 


3 


8 


20 


333 


591 


339 


649 


884 


580 


695 


799 


493 


3 


9 1 


21 


403 


623 


409 


683 


983 


611 


722 


841 


517 


3 


9 


22 


473 


654 


480 


718 


82 


641 


769 


883 


545 


3 


10 


23 


543 


685 


550 


752 


182 


672 


805 


925 


571 


3 


10 


24 


614 


716 


621 


786 


281 


702 


842 


967 


597 


3 


11 


25 


684 


747 


691 


820 


389 


733 


878 


9 


623 


4 


11 


26 


754 


778 


762 


854 


479 


763 


915 


52 


649 


4 


11 


27 


824 


809 


832 


888 


578 


794 


952 


94 


675 


4 


12 


28 


894 


840 


903 


923 


677 


824 


988 


136 


701 


4 


12 


29 


964 


872 


973 


957 


777 


855 


25 


178 


727 


4 


13 


30 


34 


903 


43 


991 


876 
975 


885 


61 


220 


753 


4 


13 


31 


105 


934 


114 


25 


916 


98 


262 


779 . 


4 


14 ; 



2b 



24 



TABLE XIX. 

moon's motions for hours. 



Hours. 


1 


2 

27 


3 


4 


5 


6 


7 


8 


9 


1 


1 


43 


12 


14 


16 


2 


16 


1 


2 


2 


54 


87 


24 


28 


31 


5 


33 


2 


3 


3 


81 


130 


36 


42 


47 


7 


49 


3 


4 


5 


108 


173 


48 


56 


62 


10 


65 


4 


5 


6 


135 


217 


60 


70 


78 


12 


81 


5 


6 


7 


162 


260 


72 


84 


93 


14 


98 


6 


7 


8 


190 


303 


84 


98 


109 


17 


114 


7 


8 


9 


217 


347 


96 


112 


124 


19 


130 


8 


9 


10 


244 


390 


J 08 


126 


140 


22 


146 


9 


10 


11 


271 


433 


120 


140 


155 


24 


163 


10 


11 


12 


298 


477 


131 


154 


171 


26 


179 


11 


12 


14 


325 


520 


143 


168 


186 


29 


195 


12 


13 


15 


352 


563 


155 


182 


202 


31 


211 


13 


14 


16 


379 


607 


167 


196 


217 


34 


228 


14 


15 


17 


406 


650 


179 


210 


233 


36 


244 


15 


16 


18 


433 


693 


191 


224 


248 


38 


260 


16 


17 


19 


460 


737 


203 


238 


264 


41 


276 


17 


18 


20 


487 


780 


215 


252 


279 


43 


293 


18 


19 


22 


515 


823 


227 


266 


295 


46 


309 


19 


20 


23 


542 


867 


239 


280 


310 


48 


325 


20 


21 


24 


569 


910 


251 


294 


326 


50 


341 


21 


22 


25 


596 


953 


263 


308 


341 


53 


358 


22 


33 


26 


623 


997 


275 


322 


357 


55 


374 


23 


24 


27 


650 


1040 


287 


336 


372 


58 


390 


24 



TABLE XIX. 

moon's motions for minutes. 



Min. 


1 




2 




3 
1 


4 



5 


6 


7 



8 



9 



10 



11 


12 


13 



u 




1 














5 





2 


4 


1 


1 


1 





1 




















10 





5 


7 


2 


2 


3 





3 

















1 


15 





7 


11 


3 


3 


4 




4 





1 





1 





1 


20 





9 


14 


4 


5 


5 




5 





1 





1 





1 


25 





11 


18 


5 


6 


6 




7 





1 




1 




2 


30 


1 


14 


22 


6 


7 


8 




8 





1 


1 


1 




2 


35 




16 


25 


7 


8 


9 




10 




2 




2 




2 


40 




18 


29 


8 


9 


10 


2 


11 




2 




2 




3 


45 




20 


32 


9 


10 


12 


2 


12 




2 




2 




3 


50 




23 


36 


10 


11 


13 


2 


13 


1 


2 




2 




3 


55 




25 


40 


11 


13 


14 


2 


15 




3 




3 




4 


60 ] 




27 


43 


12 


14 


15 


2 


16 




3 




3 




4 



TABLES. 



25 



HELIOCENTRIC LONGITUDES, ETC. OF THE PLANET VENUS, AT THE TIMES OF 

THE NEXT TWO TRANSITS OVER THE SUN's DISC. 

The subject matter of this table is connected with Chapter IX, page 119. 



Times. 


Hel. Long, from 
true Equinox. 


Hel. Lat. 


Rad. Vec. 


1874, Dec. 8th, at 12h. 
I6h. 
20h. 

1882, Dec. 6th, at noon. 
4h. 
8h. 


76° 41' 36.6" 

76 57 44.1 

77 13 51.5 

74 12 55.7 
74 29 2.5 
74 45 9.7 


4' 6.3" N. 

5 3.5 

6 1.0 

4 58.1 S. 
4 0.8 
3 3.5 


0.7203632 
0.7203449 
0.7203268 

0.7205502 
0.7205315 
0.7205127 



DIP OF THE HORIZON. 
For the principle of computing the dip of the horizon see text- note, 



54. 



Hight 




Hight 






in 


Dip. 


m 


Dip 


feet. 




feet. 






1 


1' 1" 


16 


4' 


3" 


2 


1 26 


17 


4 


11 


3 


1 45 


18 


4 


18 


4 


2 2 


19 


4 


25 


5 


2 16 


20 


4 


32 


6 


2 29 


21 


4 


39 


7 


2 41 


22 


4 


45 


8 


2 52 


23 


4 


52 


9 


3 2 


24 


4 


58 


10 


3 12 


25 


5 


4 


11 


3 22 


26 


5 


10 


12 


3 31 


28 


5 


22 


13 


3 39 


30 


5 


33 


14 


3 48 


35 


6 


1 


15 


3 56 


40 


6 


25 



SUN S SEMIDIAMETER FOR EVERY TENTH DAT OF THE YEAR. 



Days. 

1 
11 
21 


Jan. 
/ // 
16 18 
16 17 
16 17 


July. 

/ // 
15 46 
15 46 
15 46 


Days. 

1 
11 

21 


April. 
/ // 
16 1 
15 58 
15 55 


Oct. 

/ // 
16 1 
16 3 
16 7 


1 

11 
21 


Feb. 
16 15 
16 13 
16 11 


August. 
15 47 
15 49 
15 51 


1 
11 

21 


May. 
15 53 
15 51 
15 49 


Nov. 
16 9 
16 12 
16 14 


1 
11 

21 


March. 

16 10 

16 7 

1 16 4 


Sept. 
15 53 
15 56 

15 58 


1 
11 

J 21 


June. 
15 48 
15 46 
15 46 


Dec. 
16 16 
16 17 

16 18 



22 



26 




TABLE XX. 










moon's epochs. 








Years. 


Evection. 


Anomaly. 


Variation. 


Longitude. 




s o ' " 


s ° ' " 


s 


o 


/ // 


s O ' » 


1846 


2 45 6 


26 21 2 


1 


5 


48 4 


10 15 48 23 


1847 


7 21 16 35 


3 25 4 23 


5 


15 


25 29 


2 25 11 28 


1848 B. 


1 23 7 5 


7 6 51 37 


10 


7 


14 21 


7 17 45 8 


1849 


7 13 38 35 


10 5 34 57 


2 


16 


51 46 


11 27 8 14 


1850 


1 4 10 4 


1 4 18 18 


6 


26 


29 11 


4 6 31 20 


1851 


6 24 41 35 


4 3 1 38 


11 


6 


6 36 


8 15 54 25 


1852 B. 


26 32 5 


7 14 48 53 


3 


27 


55 29 


1 8 28 6 


1853 


6 17 3 34 


10 13 32 13 


8 


7 


32 53 


5 17 51 11 


1854 


7 35 4 


1 12 15 34 





17 


10 19 


9 27 14 17 


1855 


5 28 6 33 


4 10 58 54 


4 


26 


47 43 


2 6 37 . 22 


1856 B. 


11 29 57 3 


7 22 46 9 


9 


18 


36 36 


6 29 11 3 


1857 


5 20 28 33 


10 21 29 29 


1 


28 


14 1 


11 8 34 9 


j 1858 


11 11 2 


1 20 12 50 


6 


7 


51 26 


3 17 57 14 


1859 


5 1 31 33 


4 18 56 10 


10 


17 


28 52 


7 27 20 20 


j 1860 B, 


11 3 22 3 


8 43 25 


3 


9 


17 44 


19 54 


i 1861 


4 23 53 33 


10 29 26 45 


7 


18 


55 9 


4 29 17 6 


1 1862 


10 14 25 3 


1 28 10 6 


11 


28 


32 34 


9 8 40 12 


| 1863 


4 4 56 33 


4 26 53 27 


4 


8 


10 


1 18 3 18 


1864 B. 


10 6 47 2 


8 8 40 41 


8 


29 


58 51 


6 10 36 58 


1865 


3 27 18 32 


11 7 24 2 


1 


9 


36 17 


10 20 4 


1866 


9 17 50 2 


2 6 7 23 


5 


19 


13 42 


2 29 23 10 


1867 


3 8 21 32 


5 4 50 43 


9 


28 


51 8 


7 8 46 15 


1868 B. 


9 10 12 2 


8 16 37 58 


2 


20 


40 


1 19 56 


1869 


3 43 33 


11 15 21 19 


7 





17 25 


4 10 43 2 


1870 


8 21 15 2 


2 14 4 40 


11 


9 


54 50 


8 20 6 8 


1871 


2 11 45 31 


5 12 47 1 


3 


19 


31 16 


29 28 13.7 


1872 B. 


8 2 17 


8 11 30 21.7 


7 


29 


8 41 


5 8 51 19.4 


1873 


2 4 7 31 


11 23 17 36.6 





20 


57 36 


10 1 25 0.3 


1874 


7 24 39 


2 22 57.3 


5 





35 


2 10 48 6 


1875 


1 15 10 29 


5 20 44 18 


9 


10 


12 24 


6 20 11 11.7 


1876 B. 


7 5 41 59 


8 19 27 38.7 


1 


19 


49 50 


10 29 34 17.4 


1877 


1 7 32 30 


1 14 53.6 


6 


11 


38 40 


3 22 7 58.3 


1878 


6 28 3 59 


2 29 58 14.3 


10 


21 


16 5 


8 1 31 4 


1879 


18 35 28 


5 28 41 35 


3 





53 30 


10 54 9.7 


1880 B. 


6 9 6 58 


8 27 24 55.7 


7 


10 


30 55 


4 20 17 15.4 


1881 


10 57 29 


9 12 10.6 





2 


19 47 


9 12 50 56.3 


1882 


6 1 28 58 


3 7 55 31.3 


4 


11 


57 12 


1 22 14 2.0 


1883 


11 22 27 


6 6 38 52.0 


8 


21 


34 37 


6 1 37 7.7 


1884 B. 


5 12 31 56 


9 5 22 12.7 


1 


1 


12 2 


10 11 13.4 


1885 


11 14 22 28 


17 9 27.6 


5 


23 


54 


3 3 33 54.3 


1886 


5 4 53 57 


3 15 52 48.3 


10 


2 


38 19 


7 12 57 0.0 


1887 


10 25 25 26 


6 14 36 9.0 


2 


12 


15 44 


11 22 20 5.7 


1888 B. 


4 15 56 57 


9 13 19 29.7 


6 


21 


53 9 


4 1 43 11.0 


1889 


10 17 47 28 


25 6 44.6 


11 


13 


42 1 


8 24 16 51.9 


1890 


4 8 18 57 


3 23 50 5.3 


3 


23 


19 26 


1 3 39 57.6 



TABLE X . 

moon's epochs. 



27 



Years. 


Supp. of Node. 


II. 


V. 


VI. 


VII. 


VIII. 


IX. 


i 
X. 


1846 


s 
4 


o 
16 


35 


9 


s 
11 


o 
7 


i 
56 


254 


258 


937 


941 


847 


| 
113 


1847 


5 


5 


54 


52 


2 


28 


38 


668 


670 


245 


247 


927 


053 


1848 B. 


2 


25 


17 


45 


7 





9 


116 


122 


582 


587 


042 


997 


1849 


6 


14 


37 


27 


10 


20 


41 


531 


535 


889 


893 


122 


937 


1 1850 


7 


3 


57 


9 


2 


11 


13 


944 


947 


196 


200 


202 


876 1 


1851 


7 


23 


16 


51 


6 


1 


45 


358 


359 


504 


506 


282 


816 ! 


1852 B. 


8 


12 


39 


44 


10 


3 


27 


806 


811 


841 


846 


398 


760 : 


1853 


9 


1 


59 


26 


1 


23 


59 


220 


223 


148 


152 


477 


700 ; 


1854 


9 


21 


19 


9 


5 


14 


31 


634 


636 


456 


459 


557 


639 ! 


1855 


10 


10 


38 


51 


9 


5 


3 


047 


048 


763 


765 


637 


579 I 


1856 B. 


11 





1 


44 


1 


6 


44 


495 


500 


100 


105 


753 


523 i 


1857 


11 


19 


21 


26 


4 


27 


16 


909 


912 


407 


411 


832 


463 ! 


1858 





8 


41 


8 


8 


17 


48 


323 


325 


715 


718 


912 


402 ! 


1859 





28 





51 





8 


20 


736 


737 


023 


024 


992 


342 ! 


1860 B. 


1 


17 


23 


43 


4 


10 


1 


184 


189 


359 


364 


108 


286 | 


1861 


2 


6 


43 


27 


8 





33 


598 


601 


666 


670 


187 


226 


1862 


2 


26 


3 


9 


11 


21 


5 


012 


014 


974 


977 


267 


165 


1863 


3 


15 


23 


11 


3 


11 


37 


426 


426 


282 


283 


347 


105 J 


1864 B. 


4 


4 


45 


44 


7 


13 


18 


873 


878 


618 


623 


463 


049 


1865 


4 


24 


5 


46 


11 


3 


50 


287 


291 


926 


929 


542 


989 1 


1866 


5 


13 


25 


28 


2 


24 


22 


701 


703 


233 


236 


622 


928 


1867 


6 


2 


45 


10 


6 


14 


54 


115 


115 


544 


542 


702 


868 


1868 B. 


6 


22 


7 


43 


10 


16 


36 


563 


567 


877 


882 


818 


812 


1869 


7 


11 


27 


46 


2 


7 


8 


977 


980 


185 


188 


897 


752 


1870 


8 





47 


28 


5 


27 


40 


390 


392 


493 


495 


977 


691 


1871 


8 


20 


6 


49 


9 


18 


11 


803 


804 


600 


802 


057 


630 


1872 B. 


9 


9 


26 


31 


1 


8 


43 


216 


216 


108 


110 


137 


569 


1873 


9 


28 


49 


24 


5 


10 


25 


664 


668 


444 


450 


252 


514 


i 1874 


10 


18 


9 


6 


9 





57 


077 


080 


752 


758 


332 


453 


1875 


11 


7 


28 


48 





21 


29 


490 


492 


054 


064 


412 


392 

] 


1876 B. 


11 


26 


48 


31 


4 


12 


1 


904 


905 


364 


370 


492 


331 i 


1877 





16 


11 


24 


8 


13 


42 


352 


357 


700 


710 


607 


276 


1878 


1 


5 


31 


6 





4 


14 


765 


769 


008 


018 


687 


215 


1879 


] 


24 


50 


48 


3 


24 


46 


178 


181 


316 


326 


767 


154 i 


1880 B. 


2 


14 


10 


30 


7 


15 


18 


593 


593 


624 


630 


847 


093 1 

i 


1881 


3 


3 


33 


23 


11 


16 


59 


041 


045 


960 


970 


962 


038 


1882 


3 


22 


53 


5 


3 


7 


31 


454 


457 


268 


278 


042 


977 


1883 


4 


12 


12 


47 


6 


28 


3 


867 


869 


576 


586 


122 


916 


1884 B. 


5 


1 


32 


29 


10 


18 


35 


280 


281 


884 


894 


202 


855 


1885 


5 


20 


55 


22 


3 


20 


16 


728 


733 


220 


234 


317 


800 


1886 


6 


10 


15 


4 


6 


10 


48 


141 


145 


528 


542 


397 


739 


1887 


6 


29 


34 


46 


10 


1 


20 


554 


557 


836 


850 


477 


678 


1888 B. 


7 


18 


54 


28 


1 


21 


52 


967 


969 


144 


158 


557 


617 


1889 


8 


8 


17 


21 


5 


23 


33 


415 


421 


480 


498 


672 


562 


, 1890 


8 


27 


36 


3 


9 


14 


5 


828 


833 


788 | 


806 


752 


501 j 



2b* 



28 



TABLE XX. 
moon's motions for months. 



Months. 


FiVection 




Anomaly 




Variation. 


M. 


Lor 


gitude. 




s 


o 


/ 


// 


s 


o 


' 


a 


B 


o 


/ // 


s 


o 




n 


T ~i Com.. 
Jan ' I Bis. . 















































11 


18 


41 


1 


11 


16 


56 


6 


11 


17 


48 33 


11 


16 


49 


25 


Feb.]£ om " 
J Bis. . 


11 


20 


48 


42 


1 


15 





53 





17 


54 48 


1 


18 


28 


6 


11 


9 


29 


43 


1 


1 


56 


59 





5 


43 21 


1 


5 


17 


31 




10 


7 


40 


26 


1 


20 


50 


4 


11 


29 


15 15 


1 


27 


24 


27 


April 


9 


28 


29 


8 


3 


5 


50 


57 





17 


10 3 


3 


15 


52 


32 


May 


9 


7 


58 


51 


4 


7 


47 


56 





22 


53 24 


4 


21 


10 


3 




8 


28 


47 


33 


5 


22 


48 


49 


1 


10 


48 11 


6 


9 


38 


9 




8 


8 


17 


16 


6 


24 


45 


48 


1 


16 


31 32 


7 


14 


55 


40 




17 
i 


29 


5 


59 


8 


9 


46 


42 


2 


4 


26 20 


9 


3 


23 


46 


September.. . 


7 


19 


54 


41 


9 


24 


47 


35 


2 


22 


21 7 


10 


21 


51 


52 




6 


29 


24 


24 


10 


26 


44 


34 


2 


28 


4 28 


11 


27 


9 


22 


November. . . 


6 


20 


13 


6 





11 


45 


27 


3 


15 


59 16 


1 


15 


37 


28 


December . . . 


5 


29 


42 


49 


1 


13 


42 


26 


3 


21 


42 37 


2 


20 


54 


59 



TABLE XX. 

moon's motions for days. 



Days. 


Evection. 


Anomaly. 


Variation. 


Mean Longitude. 


1 


0s 


19 0' 0" 


0s 


0° 0' 


0" 


0s 


0° 


0' 


0" 


0s 0° 


0' 


0" 


2 





11 18 59 





13 3 


54 





12 


11 


27 


13 


10 


35 


3 





22 37 59 





26 7 


48 





24 


22 


53 


26 


21 


10 


4 


1 


3 56 58 


1 


9 11 


42 


1 


6 


34 


20 


1 9 


31 


45 


5 


1 


15 15 58 


1 


22 15 


36 


1 


18 


45 


47 


1 22 


42 


20 


6 


1 


26 34 57 


2 


5 19 


30 


2 





57 


13 


2 5 


52 


55 


7 


2 


7 53 57 


2 


18 23 


24 


2 


13 


8 


40 


2 19 


3 


30 


8 


2 


19 12 56 


3 


1 27 


18 


2 


25 


20 


7 


3 2 


14 


5 


9 


3 


31 55 


3 


14 31 


12 


3 


7 


31 


34 


3 15 


24 


40 


10 


3 


11 50 55 


3 


27 35 


6 


3 


19 


43 





3 28 


35 


15 


11 


3 


23 9 54 


4 


10 39 





4 


1 


54 


27 


4 11 


45 


50 


12 


4 


4 28 54 


4 


23 42 


54 


4 


14 


5 


54 


4 24 


56 


25 


13 


4 


15 47 53 


5 


6 46 


48 


4 


26 


17 


20 


5 8 


7 





14 


4 


27 6 53 


5 


19 50 


42 


5 


8 


28 


47 


5 21 


17 


35 


15 


5 


8 25 52 


6 


2 54 


36 


5 


20 


40 


14 


6 4 


28 


10 


16 





19 44 51 


6 


15 58 


29 


6 


2 


51 


40 


6 17 


38 


45 


17 


6 


1 3 51 


6 


o 


23 


6 


15 


3 


7 


7 


49 


20 


18 


6 


12 22 50 


7 


12 6 


17 


6 


27 


14 


34 


7 13 


59 


55 


19 


6 


23 41 50 


7 


25 10 


11 


7 


9 


26 


1 


7 27 


10 


30 


20 


7 


5 49 


8 


8 14 


5 


7 


21 


37 


27 


8 10 


21 


5 


21 


7 


16 19 49 


8 


21 17 


59 


8 


3 


48 


54 


8 23 


31 


40 


22 


7 


27 38 48 


9 


4 21 


53 


8 


16 





21 


9 6 


42 


16 


23 


8 


8 57 47 


9 


17 25 


47 


8 


28 


11 


47 


9 19 


52 


51 


24 


8 


20 16 47 


10 


29 


41 


9 


10 


23 


14 


10 3 


3 


26 


25 


9 


1 35 46 


10 


13 33 


35 


9 


22 


34 


41 


10 16 


14 


1 


26 


9 


12 54 46 


10 


26 37 


29 


10 


4 


46 


7 


10 29 


24 


36 


27 


9 


24 13 45 


11 


9 41 


23 


10 


16 


57 


34 


11 12 


35 


11 


28 


10 


5 32 45 


11 


22 45 


17 


10 


29 


9 


1 


11 25 


45 


46 


29 


10 


16 51 44 





5 49 


11 


11 


11 


20 


28 


8 


56 


21 


30 


10 


28 10 43 





18 53 


5 


11 


23 


31 


54 


22 


6 


56 


31 


11 


9 29 43 


1 


1 56 


59 


o 


5 


43 


21 


1 5 


17 


31 



TABLE XX. 
moon's motions for months. 



29 



Months. 


Supp. of Nc 


de. 


II. 


V. 


VI. 


VII. 


VIII. 

000 

966 


IX. 

000 
964 


X. 


T "1 Com.. 
Jan -jBis. . 


s 



11 


o 



29 



56 




49 


s 



11 


o 


18 



51 


000 
966 


000 
961 


000 

972 


000 
995 


■Feb.lS-° m " 
J Bis. . 





1 


38 


30 


11 


15 


43 


54 


224 


875 


45 


111 


165 





1 


35 


19 


11 


4 


34 


20 


185 


847 


11 


75 


159 







3 


7 


27 


9 


27 


59 


7 


330 


666 


989 


114 


313 







4 


45 


57 


9 


13 


42 


61 


554 


542 


34 


225 


478 


May 





6 


21 


16 


8 


18 


15 


81 


738 


389 


46 


300 


638 







7 


59 


46 


8 


3 


58 


136 


962 


264 


91 


411 


802 


July 





9 


35 


5 


7 


8 


32 


156 


147 


112 


103 


486 


962 







11 


13 


35 


6 


24 


15 


210 


371 


987 


147 


497 


126 


September.. . 





12 


52 


5 


6 


9 


58 


265 


595 


862 


193 


708 


291 







14 


27 


24 


5 


14 


32 


285 


780 


710 


204 


783 


451 


November. . . 





16 


5 


53 


5 





15 


339 


4 


585 


250 


894 


615 


December . . . 





17 


41 


13 


4 


4 


49 


359 


188 


|432 


261 


969 


775 



TABLE XX. 

moon's motions foe days. 



Days. 



1 

2 

3 

4 

5 

6 

8 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 



Supp. of Node 



0° 





































1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 



0' 0" 

3 11 

6 21 

9 32 

12 52 
15 53 
19 4 
22 14 
25 25 
28 36 
31 46 

34 57 
38 7 
41 18 
44 29 
47 39 
50 50 
54 1 
57 11 

22 

3 33 

6 43 

9 54 

13 5 
16 15 
19 26 
22 36 
25 47 
28 58 
32 8 

35 19 



II. 



0s 0o 0' 

11 9 

22 18 

3 27 

14 37 

25 46 

6 55 

2 18 4 

2 29 13 

10 22 

21 31 

2 40 

13 50 

24 59 

6 8 

17 17 

28 26 

9 35 

20 44 

1 53 

13 3 

24 12 

5 21 

16 30 

27 39 

8 48 

19 57 

1 6 

10 12 16 

10 23 25 

11 4 34 



9 

9 

10 



000 
34 
68 

102 

136 

170 

204 

238 

272 

306 

340 

374 

408 

442 

476 

510 

544 

578 

612 

646 

680 

714 

748 

782 

816 

850 

884 

918 

952 

986 

020 



VI. 



000 
39 
79 
118 
158 
197 
237 
276 
316 
355 
395 
434 
474 
513 
553 
592 
632 
671 
711 
750 
790 
829 
869 
908 
948 
987 
027 
066 
106 
145 
185 



VII. 



000 
28 
56 
85 
113 
141 
169 
198 
226 
254 
282 
311 
339 
367 
395 
424 
452 
480 
508 
537 
565 
593 
621 
650 
678 
706 
734 
762 
791 
819 
847 



VIII. 



000 
34 
67 
101 
135 
169 
202 
236 
270 
303 
337 
371 
405 
438 
472 
506 
539 
573 
607 
641 
674 
708 
742 
775 
809 
843 
877 
910 
944 
978 
011 



IX. 



000 
36 

72 
108 
143 
179 
215 
251 
287 
323 
358 
394 
43Q 
466 
502 
538 
573 
609 
645 
681 
717 
753 
788 
824 
860 
896 
932 
968 
003 
039 
075 



X. 



000 

5 

11 

16 

21 

27 

32 

37 

43 

48 

53 

58 

64 

69 

74 

80 

85 

90 

96 

101 

106 

112 

117 

122 

128 

133 

138 

143 

149 

154 

151 



m 



TABLE XX. 

moon's motions for hours. 



Hours. 


Evection. 


Anomaly. 


Variati* 


311 


Longitude. 


o 


i 


a 


o 


/ 


a 


o 


i 


ll 


o 


i a 


1 





28 


17 





32 


40 





30 


29 





32 56 


2 





56 


35 


1 


5 


19 


1 





57 


1 


5 53 


3 


1 


24 


52 


1 


37 


59 


1 


31 


26 


1 


38 49 


4 


1 


53 


10 


2 


10 


39 


2 


1 


54 


2 


11 46 


5 


2 


21 


27 


2 


43 


19 


2 


32 


23 


2 


44 42 


6 


2 


49 


45 


3 


15 


58 


3 


2 


52 


3 


17 39 


7 


3 


18 


2 


3 


48 


38 


3 


33 


20 


3 


50 35 


8 


3 


46 


20 


4 


21 


18 


4 


3 


49 


4 


23 32 


9 


4 


14 


37 


4 


53 


58 


4 


34 


17 


4 


56 28 


10 


4 


42 


55 


5 


26 


37 


5 


4 


46 


5 


29 25 


11 


5 


11 


12 


5 


59 


17 


5 


35 


15 


6 


2 21 


12 


5 


39 


30 


6 


31 


57 


6 


5 


43 


6 


35 17 


13 


6 


7 


47 


7 


4 


37 


6 


36 


12 


7 


8 14 


14 


6 


36 


5 


7 


37 


16 


7 


6 


40 


7 


41 10 


15 


7 


4 


22 


8 


9 


56 


7 


37 


9 


8 


14 7 


! 16 


7 


32 


40 


8 


42 


36 


8 


7 


38 


8 


47 3 


17 


8 





57 


8 


15 


16 


8 


38 


6 


9 


20 


18 


8 


29 


15 


9 


47 


55 


9 


8 


35 


9 


52 56 


19 


8 


57 


32 


10 


20 


35 


9 


39 


3 


10 


25 53 


20 


9 


25 


50 


10 


53 


15 


10 


9 


32 


10 


58 49 


21 


9 


54 


7 


11 


25 


55 


10 


40 


1 


11 


31 46 


22 


10 


22 


24 


11 


58 


34 


11 


10 


29 


12 


4 42 


23 


10 


50 


42 


12 


31 


14 


11 


40 


58 


12 


37 39 


24 

4 


11 


18 


59 


13 


3 


54 


12 


11 


27 


13 


10 35 



TABLE XXI. 

moon's motions for minutes. 



Min. 


Evec. 


Anomaly. 


Variations. 


Longitude. 


Sup. 
Node. 


II. 




/ 


// 


/ // 


/ // 


/ 


a 


a 


// 


1 





28 


33 


30 





33 








5 


2 


21 


2 43 


2 32 


2 


45 


1 


2 


10 


4 


43 


5 27 


5 5 


5 


29 


1 


5 


15 


7 


4 


8 10 


7 37 


8 


14 


2 


7 


20 


9 


26 


10 53 


10 10 


10 


59 


3 


9 


25 


11 


47 


13 37 


12 42 


13 


43 


3 


12 


30 


14 


9 


16 20 


15 14 


16 


28 


4 


14 


35 


16 


30 


19 3 


17 47 


19 


13 


5 


16 


40 


18 


52 


21 46 


20 19 


21 


58 


5 


19 


45 


21 


13 


24 30 


22 52 


24 


42 


6 


21 


50 


23 


34 


27 13 


25 24 


27 


27 


7 


23 


55 


25 


56 


29 56 


27 56 


30 


12 


7 


26 


60 


28 


17 


32 40 


30 29 


32 


56 


8 


28 







TABLE 


XX. 














moon's motions 


> FOR 


HOURS. 








r ' 
Hours. 


Supp. of 
Node. 


II. 


V. 


VI. 


VII. 


VIII. 


IX. 


X. 




/ // 


O ' 














1 


8 


28 


1 


2 


1 


1 


1 





2 


16 


56 


3 


3 


2 


3 


3 





3 


24 


1 24 


4 


5 


4 


4 


4 


1 


4 


32 


1 52 


6 


V 


5 


6 


6 


1 


5 


40 


2 19 


7 


8 


6 


7 


7 


1 


6 


48 


2 47 


9 


10 


7 


9 


9 


1 


7 


56 


3 15 


10 


12 


8 


10 


10 


2 


8 


1 4 


3 43 


11 


13 


9 


11 


12 


2 


9 


1 11 


4 11 


13 


15 


11 


13 


13 


2 


10 


1 19 


4 39 


14 


16 


12 


14 


15 


2 


31 


1 27 


5 7 


16 


18 


13 


15 


16 


2 


12 


1 35 


5 35 


17 


20 


14 


17 


18 


3 


13 


1 43 


6 2 


18 


21 


15 


18 


19 


3 


14 


1 51 


6 30 


20 


23 


16 


19 


21 


3 


15 


1 59 


6 58 


21 


25 


18 


21 


22 


3 


16 


2 7 


7 26 


23 


26 


19 


22 


24 


4 


17 


2 15 


7 54 


24 


28 


20 


24 


25 


4 


18 


2 23 


8 22 


26 


29 


21 


25 


27 


4 


19 


2 31 


8 50 


27 


31 


22 


27 


28 


4 


20 


2 39 


9 18 


28 


32 


24 


28 


30 


4 


21 


2 47 


9 45 


30 


34 


25 


29 


31 


5 


22 


2 55 


10 13 


31 


36 


26 


31 


33 


5 


23 


3 3 


10 41 


33 


38 


27 


32 


34 


5 


24 


3 11 


11 9 


34 


39 


28 


34 


36 


5 



















31 



TABLE A.* 

PERTURBATIONS OF EARTH'S 
RADIUS VECTOR. 



TABLE B. 

f)'s APPROX. LAT. ARG. N. 



Arg. 



I. 

8 


II. 
4 


III. 
3 


Arg. 


I. 


II. 


III. 


500 


2 





4 


50 


8 


4 


3 


550 


2 


1 


4 


100 


7 


4 


2 


600 


3 


1 


3 


150 


7 


4 


1 


650 


3 


2 


2 


200 


6 


4 





700 


4 


3 


1 


250 


5 


4 





750 


5 


4 





300 


4 


3 


1 


800 


6 


4 





350 


3 


2 


2 


850 


7 


4 


1 


400 


3 


1 


3 


900 




4 


2 


450 


2 


1 


4 


950 


8 


4 


3 


500 


2 





4 


1000 


8 


4 


3 



N. 


N. 


S. 


S. 


O's 


A. 



D. 


D. 


A. 


Lat. 


500 


500 


1000 





5 


495 


505 


995 


9 41 


10 


490 


510 


990 


19 22 


15 


485 


515 


985 


29 3 


20 


480 


520 


980 


38 40 


25 


475 


525 


975 


48 18 


30 


470 


530 


970 


58 40 


35 


465 


535 


965 


67 28 


40 


460 


540 


960 


76 45 


45 


455 


545 


955 


86 21 


50 


450 


550 


950 


95 26 j 


55 


445 


555 


945 


04 56 ! 



Tables A. and B. are put in this place on account of the convenience in the page. 



32 TABLE XXI 

FIRST EQUATION OP MOON's LONGITUDE. — ARGUMENT 1. 



IT 



Arg. 




100 
200 
300 
400 
500 
600 
700 
800 
900 
1000 
1100 
1200 
1300 
1400 
1500 
1600 
1700 
1800 
1900 
2000 
2100 
2200 
2300 
2400 
2500 
2600 
2700 
2800 
2900 
3000 
3100 
3200 
3300 
3400 
3500 
3600 
3700 
3800 
3900 
4000 
4100 
4200 
4300 
4400 
4500 
4600 
4700 
4800 
4900 
5000 



12 40 

11 58 

11 16 

10 34 



9 

9 

10 

11 

11 



53 
12 



8 32 



54 
16 
40 

6 
33 

2 
32 

5 
40 
17 
56 
38 
22 

9 
58 
50 
44 
41 
41 
43 
48 
55 

5 
17 
32 
49 

8 
30 
53 
19 
46 
16 
47 
19 
53 
28 

5 
42 
20 
59 
39 
19 
59 
12 40 



Diff. 



42 

42 

42 

41 

41 

40 

38 

38 

36 

34 

33 

31 

30 

27 

25 

23 

21 

18 

16 

13 

11 

8 

6 

3 



2 

5 

7 

10 
12 
15 
17 
19 
22 
23 
26 
27 
30 
31 
32 
34 
35 
37 
37 
38 
39 
40 
40 
40 
41 



Arg 



5000 
5100 
5200 
5300 
5400 
5500 
5600 
5700 
5800 
5900 
6000 
6100 
6200 
6300 
6400 
6500 
6*00 
6700 
6800 
6900 
7000 
7100 
7200 
7300 
7400 
7500 
7600 
7700 
7800 
7900 
8000 
8100 
8200 
8300 
8400 
8500 
8600 
8700 
8800 
8900 
9000 
9100 
9200 
9300 
9400 
9500 
9600 
9700 
9800 
9900 
10000 



12 40 

13 20 

14 1 

14 41 

15 20 

16 

16 38 

17 15 

17 52 

18 27 

19 1 

19 33 

20 4 

20 33 

21 1 
21 27 

21 50 

22 12 
22 31 

22 48 

23 3 
23 15 
23 25 
23 32 
23 37 
23 39 
23 39 
23 36 
23 30 
23 22 
23 11 
22 58 
22 42 
22 24 
22 3 
21 40 
21 15 
20 48 
20 18 
19 47 
19 14 
18 40 
18 4 
17 26 
16 48 
16 8 
15 27 
14 46 
14 4 
13 22 
12 40 



Diff. 



40 
41 
40 
39 
40 
38 
37 
37 
35 
34 
32 
31 
29 
28 
26 
23 
22 
19 
17 
15 
12 
10 
7 
5 
2 

3 
6 
8 
11 
13 
16 
18 
21 
23 
25 
27 
30 
31 
33 
34 
36 
38 
38 
40 
41 
41 
42 
42 
42 



TABLE XXII. 

EQUATIONS 2 TO 7 OF MOOn's LONGITUDE. 



33 



-ARGUMENTS 2 TO 



i . 



Arg. 


2 


3 


4 


5 


6 


7 


Arg. 




, 


II 


, 


II 


i 


II 


/ 


II 


/ 


a 


/ 


II 




2500 


4 


57 





2 


6 


30 


3 


39 





6 





1 


2500 


2600 


4 


57 





2 


6 


30 


3 


39 





6 





1 


2400 


2700 


4 


56 





3 


6 


29 


3 


38 





7 





1 


2300 


2800 


4 


55 





3 


6 


27 


3 


37 





8 





2 


2200 


2900 


4 


53 





4 


6 


24 


3 


36 





9 





3 


2100 


3000 


4 


50 





5 


6 


21 


3 


34 





10 





4 


2000 


3100 


4 


47 





6 


6 


17 


3 


32 





12 





5 


1900 


3200 


4 


43 





8 


6 


12 


3 


29 





14 





6 


1800 


3300 


4 


39 





9 


6 


7 


3 


26 





17 





8 


1700 


3400 


4 


34 





11 


6 


1 


3 


22 





19 





10 


1600 


3500 


4 


29 





13 


5 


54 


3 


18 





22 





12 


1500 


3600 


4 


23 





15 


5 


47 


3 


14 





25 





14 


1400 


3700 


4 


17 





18 


5 


39 


3 


10 





29 





17 


1300 


3800 


4 


11 





20 


5 


30 


3 


5 





33 





19 


1200 


3900 


4 


4 





23 


5 


21 


3 








37 





22 


1100 


4000 


3 


57 





26 


5 


12 


2 


54 





41 





25 


1000 


4100 


3 


49 





29 


5 


2 


2 


49 





45 





28 


900 


4200 


3 


41 





32 


4 


52 


2 


43 





50 





31 


800 


4300 


3 


33 





35 


4 


41 


2 


37 





54 





35 


700 


4400 


3 


24 





39 


4 


30 


2 


30 





59 





38 


600 


4500 


3 


15 





42 


4 


19 


2 


24 


1 


4 





42 


500 


4600 


3 


7 





46 


4 


7 


2 


17 


1 


9 





45 


400 


4700 


2 


58 





49 


3 


56 


2 


10 


1 


14 





49 


300 


4800 


2 


48 





53 


3 


44 


2 


4 


1 


19 





53 


200 


4900 


2 


39 





56 


3 


32 


1 


57 


1 


25 





56 


100 


5000 


2 


30 


] 





3 


20 


1 


50 


1 


30 


1 





0000 


5100 


2 


21 


1 


4 


3 


8 


1 


43 


1 


35 


1 


4 


9900 


5200 


2 


11 


1 


7 


2 


56 


1 


36 


1 


40 


1 


7 


9S00 


5300 


2 


2 


1 


11 


2 


44 


1 


29 


1 


46 


1 


11 


9700 


5400 


1 


53 


1 


14 


2 


33 


1 


23 


1 


51 


1 


15 


9600 


5500 


1 


44 


1 


18 


2 


21 


1 


16 


1 


56 


1 


18 


9500 


5600 


1 


36 


1 


21 


2 


10 


1 


10 


2 


1 


1 


22 


9400 


5700 


1 


27 


1 


25 


1 


59 


1 


3 


2 


6 


1 


25 


9300 


5800 


1 


19 


1 


28 


1 


48 





57 


2 


10 


1 


28 


9200 


5900 


1 


11 


1 


31 


1 


38 





51 


2 


15 


1 


32 


9100 


6000 


1 


3 


1 


34 


1 


28 





46 


2 


19 


1 


35' 


9000 


6100 





56 


1 


37 


1 


19 





40 


2 


23 


1 


38 


8900 


6200 





49 


1 


39 


1 


10 





35 


2 


27 


1 


40 


8800 


6300 





33 


1 


42 


1 


1 





30 


2 


31 


1 


43 


8700 


6400 





36 


1 


44 





53 





26 


2 


35 


1 


46 


8600 


6500 





31 


1 


47 





46 





21 


2 


38 


1 


48 


8500 


6600 





26 


1 


49 





39 





18 


2 


41 


1 


50 


8400 


6700 





21 


1 


51 





33 





14 


2 


43 


1 


52 


8300 


6800 





17 


1 


52 





28 





11 


2 


46 


1 


54 


8200 


6900 





13 


1 


54 





23 





8 


2 


48 


1 


55 


8100 


7000 





10 


1 


55 





19 





6 


2 


50 


1 


56 


8000 


7100 





7 


1 


56 





16 





4 


2 


51 


1 


57 


7900 


7200 





5 


• 1 


57 





13 





2 


2 


52 


1 


58 


7800 


7300 





4 


1 


57 





11 





1 


2 


53 


1 


59 


7700 


7400 





3 


1 


58 





10 





1 


2 


54 


1 


59 


7600 


7500 





3 


1 


58 





18 





1 


2 


54 


1 


59 


7500 

1 



34 TABLE XXIII. 

EQUATIONS 8 TO 9 OF MOON's LONGITUDE. ARGUMENTS 8 TO 9. 



Arg. 


8 


9 


Arg. 


8 


9 




, 


II 


/ 


II 




t 


ir 


/ ft 





1 


20 


1 


20 


5000 


1 


20 


1 SO 


100 


1 


15 


1 


29 


5100 


1 


24 


1 26 


200 


1 


11 


1 


37 


5200 


1 


29 


1 31 


300 


1 


7 


1 


46 


5300 


1 


33 


1 37 


400 


1 


2 


1 


54 


5400 


1 


37 


1 42 


500 





58 


2 


1 


5500 


1 


42 


1 47 


600 





54 


2 


8 


5600 


1 


46 


1 51 


700 





50 


2 


15 


5700 


1 


50 


1 55 


800 





46 


2 


20 


5800 


1 


54 


1 58 


900 





42 


2 


25 


5900 


1 


58 


2 


1000 





38 


2 


29 


6000 


2 


1 


2 1 


1?00 





35 


2 


32 


6100 


2 


5 


2 2 


1200 





31 


2 


34 


6200 


2 


8 


2 2 


1300 





28 


2 


35 


6300 


2 


11 


2 1 


1400 





25 


2 


35 


6400 


2 


14 


1 59 


1500 





33 


2 


34 


6500 


2 


17 


1 56 


1600 





20 


2 


32 


6600 


2 


19 


1 52 


1700 





18 


2 


29 


6700 


2 


22 


1 48 


1800 





16 





26 


6800 


2 


24 


1 43 


1900 





14 


2 


21 


6900 


2 


25 


1 38 


2000 





13 


2 


16 


7000 


2 


27 


1 32 


2100 





11 


2 


11 


7100 


2 


28 


1 25 


2200 





10 


2 


4 


7200 


2 


29 


1 18 


2300 





10 


1 


58 


7300 


2 


30 


1 11 


2400 





9 


1 


51 


7400 


2 


30 


1 4 


2500 





9 


1 


43 


7500 


2 


31 


56 


2600 





10 


1 


36 


7600 


2 


30 


49 


2700 





10 


1 


29 


7700 


2 


30 


42 


2800 





11 


1 


22 


7800 


2 


29 


36 


2900 





12 


1 


15 


7900 


2 


28 


29 


3000 





13 


1 


8 


8000 


2 


27 


24 


3100 





15 


1 


2 


8100 


2 


26 


18 


3200 





16 





57 


8200 


2 


24 


14 


3300 





18 





52 


8300 


2 


22 


10 


3400 





21 





47 


8400 


2 


20 


8 


3500 





23 





44 


8500 


2 


17 


6 


3600 





26 





41 


8600 


2 


15 


5 


3700 





29 





39 


8700 


2 


12 


5 


3800 





32 





38 


8800 


2 


9 


6 


3900 





35 





38 


8900 


2 


5 


8 


4000 





39 





39 


9000 


2 


2 


11 


4100 





42 





40 


9100 


1 


58 


15 


4200 





46 





42 


9200 


1 


54 


20 


4300 





50 





45 


9300 


1 


50 


25 


4400 





54 





49 


9400 


1 


46 


32 


4500 





58 





53 


9500 


1 


42 


39 


4600 


1 


3 





58 


9600 


1 


38 


46 


4700 


1 


7 


1 


3 


9700 


1 


33 


54 


4800 


1 


11 


1 


9 


9800 


1 


29 


1 3 


4900 


1 


16 


1 


14 


9900 


1 


24 


1 11 


5000 


1 


20 


1 


20 


10000 


1 


20 


1 20 



EQUATIONS 10 AND 11. 



TABLE XXIII. 

EQUATIONS 12 TO 19. 



35 

EQUATION 20. 



Arg. 


10 


11 


Arg. 


10 


11 




n 


n 




ii 


// 





10 


10 


500 


10 


10 


10 


9 


11 


510 


10 


11 


20 


9 


12 


520 


9 


» 


30 


8 


13 


530 


9 


12 


40 


7 


14 


540 


8 


13 


50 


7 


15 


550 


8 


14 


60 


6 


16 


560 


8 


14 


70 


6 


17 


570 


8 


15 


80 


5 


17 


580 


7 


15 


90 


5 


18 


590 


7 


15 


100 


5 


18 


600 


7 


16 


110 


4 


19 


610 


7 


Ifil 


120 


4 


19 


620 


7 


16 


130 


4 


19 


630 


7 


18 


140 


4 


19 


640 


7 


15 


150 


4 


19 


650 


8 


15 


160 


4 


19 


660 


8 


15 


170 


4 


18 


670 


8 


14 


180 


5 


18 


680 


9 


13 


190 


5 


17 


690 


9 


13 


200 


5 


16 


700 


10 


12 


210 


6 


16 


710 


10 


11 


220 


6 


15 


720 


11 


10 


230 


7 


14 


730 


11 


9 


240 


7 


13 


740 


12 


9 


250 


8 


12 


750 


12 


8 


260 


8 


11 


760 


13 


7 


270 


9 


10 


770 


13 


6 


280 


9 


10 


780 


14 


5 


290 


10 


9 


790 


14 


4 


300 


10 


8 


800 


15 


3 


310 


11 


7 


810 


15 


3 


320 


11 


6 


820 


15 


2 


330 


12 


6 


830 


16 


2 


340 


12 


5 


840 


16 


1 


350 


12 


5 


850 


16 


1 


360 


12 


5 


860 


16 


1 


370 


13 


4 


870 


16 


1 


380 


13 


4 


880 


16 


1 


390 


13 


4 


890 


16 


1 


400 


13 


4 


900 


15 


2 


410 


13 


5 


910 


15 


2 


420 


12 


5 


920 


15 


3 


430 


12 


5 


930 


14 


3 


440 


12 


6 


940 


14 


4 


450 


12 


6 


950 


13 


5 


460 


11 


7 


960 


13 


6 


470 


11 


8 


970 


12 


7 


480 


11 


8 


980 


11 


8 


490 


10 


9 


990 


11 


9 


500 


10 


10 


1000 


10 


10 



[ 
Arg.'l2 


13 


14 


lslie, 1 
1 i 


i 
17 


18 19 


Arg.! 




a 


// 




a 




n 


it 


n 




250 


2 


2 


8 





34 


3 


17 


3 


250 


260 


2 


2 


8 





34 


3 


17 


3 


240 


270 


2 


2 


8 





34 


3 


17 


3 


230 


280 


3 


2 


8 





33 


3 


17 


3 


220 


290 


3 


2 


8 





33 


4 


16 


3 


210 


300 


3 


2 


8 





33 


4 


16 


3 


200 


310 


3 


3 


9 


1 


33 ! 


4 


16 


3 


190 


320 


4 


3 


9 


1 


32 


4 


16 


4 


180 


330 


4 


4 


9 


1 


32 


4 


16 


4 


170 


340 


5 


4 


10 


2 


32 


4 


16 


4 


160 


350 


6 


5 


10 


2 


31 


5 


15 


4 


150 


360 


6 


6 


11 


2 


31 


5 


15 


5 


140 


370 


7 


7 


11 


3 


30 


5 


15 


5 


130 


380 


8 


7 


12 


3 


29 


5 


15 


5 


120 


390 


9 


8 


12 


4 


29 


6 


14 


6 


110 


400 


10 


9 


13 


4 


28 


6 


14 


6 


100 


410 


10 


10 


13 


5 


27 


6 


14 


6 


90 


420 


11 


11 


14 


5 


27 


7 


13 


7 


80 


430 


12 


12 


15 


6 


26 


7 


13 


7 


70 


440 


13 


13 


15 


6 


25 


8 


12 


7 


60 


450 


14 


14 


16 


7 


24 


8 


12 


8 


50 


460 


16 


15 


17 


7 


23 


8 


12 


8 


40 


470 


17 


16 


18 


8 


23 


9 


11 


9 


30 


480 


18 


18 


18 


9 


22 


9 


11 


9 


20 


490 


19 


19 


19 


9 


21 


10 


10 


10 


10 


500 


20 


20 


20 


10 


20 


10 


10 


10 


000 


510 


21 


21 


21 


11 


19 


10 


10 


10 


990 


520 


22 


22 


21 


11 


18 


11 


9 


11 


980 


530 


23 23 


22 


12 


17 


11 


9 


11 


970 


540 


24 


25 


23 


12 


17 


12 


8 


12 


960 


550 


25 


26 


24 


13 


16 


12 


8 


12 


950 


560 


26 


27 


24 


14115 


12 


7 


13 


940 


570 


27 


23 


25 


14 


14 


13 


7 


13 


930 


580 


28 


29 


26 


15 


13 


13 


7 


13 


920 


590 


29 


30 


26 


15 


13 


13 


6 


14 


910 


600 


30 


31 1 27 


16 


12 


14 


6 


14 


900 


610 


31 


32 


2S 


16 


11 


14 


6 


14 


890 


620 


32 


33 


25 


17 


11 


14 


5 


15 


880 


630 


33 


33 


29 


17 


10 


15 


5 


15 


870 


640 


M 


34 


,29 


18 


9 


15 


5 


15 


860 


650 134 


35 


30 


is; 9 


15 


5 


16 


850 


660 35 


36 


30 


18 8 


16 


4 


16 


840 


670 1 35 


36 


31 


19 8 


16 


4 


16 


830 


680 


36 


37 


31 


19 


8 


16 


4 


16 


820 


690 


36 


37 


31 


19 


7 


16 


4 


17 


810 


700 


37 


37 


32 


19 


7 


16 


4 


17 


800 


710 


37 


38 


32 


20 


7 


16 


3 


17 


790 


720 


37 


38 


32 


20 


6 


16 


3 


17 


780 


730 


38 


38 


32 


20 6 


16 


3 


17 


770 


740 


38 


38 


32 


20' 6 


17 


3 


17 


760 


750 


,38 


38 32 


20; 6 


17 


3 


111 


1 750 



Arg. 


20 
ii 


Arg. 





10 


500 


10 


11 


510 


20 


12 


520 


30 


13 


530 


40 


13 


540. 


50 


14 


550 


60 


15 


560 


70 


16 


570 


80 


16 


580 


90 


17 


590 


100 


17 


600 


110 


17 


610 


120 


17 


620 


130 


17 


630 


140 


17 


640 


150 


17 


650 


160 


17 


660 


170 


16 


670 


180 


16 


680 


190 


15 


690 


200 


14 


700 


2-10 


13 


710 


220 


13 


720 


230 


12 


730 


240 

i 


11 


740 


1 250 


10 


750 


1 260 


9 


760 


1 270 


8 


770 


| 280 


7 


780 


| 290 


6 


790 


300 


6 


800 


310 


5 


810 


320 


4 


820 


330 


4 


830 


340 


3 


840 


350 


3 


850 


360 


3 


860 


370 


3 


870 


380 


3 


880 


390 


3 


890 


400 


3 


900 


410 


3 


910 


420 


4 


920 


430 


4 


930 


440 


5 


940 


450 


6 


950 


460 


6 


960 


470 


7 


970 


480 


8 


980 


490 


9 


990 


500 


10 


1000 



22 



2 G 



36 



TABLE XXIV, 
Evection. Argument. — Evection Corrected. 





0s 


Is 

2° 10' 43" 


lis 


Ills 


IVs 


Vs 


Oo 


l'c 


>3Q' 0" 


2° 40 10" 


2° 50' 25" 


20 39' 8" 


2° 


9' 42" 


1 


1 


31 25 


2 11 57 


2 40 51 


2 


50 23 


2 


38 25 


2 


8 29 


2 


1 


32 51 


2 13 9 


2 41 30 


2 


50 20 


2 


37 40 


2 


7 16 


3 


1 


34 16 


2 14 21 


2 42 8 


2 


50 15 


2 


36 55 


2 


6 2 


4 


1 


35 42 


2 15 31 


2 42 45 


2 


50 9 


2 


36 8 


2 


4 47 


5 


1 


37 7 


2 16 41 


2 43 21 


2 


50 1 


2 


35 19 


2 


3 32 


6 


1 


38 32 


2 17 50 


2 43 55 


2 


49 52 


2 


34 30 


2 


2 16 


7 


1 


39 57 


2 18 58 


2 44 27 


2 


49 41 


2 


33 40 


2 


1 


8 


1 


41 21 


2 20 5 


2 44 59 


2 


49 29 


2 


32 48 


1 


59 43 


9 


1 


42 46 


2 21 11 


2 45 29 


2 


49 15 


2 


31 55 


1 


58 26 


10 


1 


44 10 


2 22 17 


2 45 57 


2 


49 


2 


31 2 


1 


57 8 


11 


1 


45 34 


2 23 21 


2 46 24 


2 


48 43 


2 


30 7 


1 


55 49 


12 


1 


46 58 


2 24 24 


2 46 50 


2 


48 26 


2 


29 11 


1 


54 30 


13 


1 


48 21 


2 25 26 


2 47 14 


2 


48 6 


2 


28 14 


1 


53 11 


14 


1 


49 44 


2 26 28 


2 47 37 


2 


47 45 


2 


27 16 


1 


51 51 


15 


1 


51 7 


2 27 28 


2 47 59 


2 


47 23 


2 


26 17 


1 


50 31 


16 


1 


52 29 


2 28 27 


2 48 19 


2 


47 


2 


25 17 


1 


49 11 


17 


1 


53 51 


2 29 25 


2 48 37 


2 


46 35 


2 


24 16 


1 


47 50 


18 


1 


55 12 


2 30 21 


2 48 54 


2 


46 8 


2 


23 14 


1 


46 29 


19 


1 


56 33 


2 31 17 


2 49 10 


2 


45 41 


2 


22 11 


1 


45 7 


20 


1 


57 53 


2 32 11 


2 49 24 


2 


45 12 


2 


21 7 


1 


43 46 


21 


1 


59 13 


2 33 5 


2 49 37 


2 


44 41 


2 


20 2 


1 


42 24 


22 


2 


32 


2 33 57 


2 49 48 


2 


44 9 


2 


18 56 


1 


41 2 


23 


2 


1 51 


2 34 48 


2 49 58 


2 


43 36 


2 


17 50 


1 


39 39 


24 


2 


3 9 


2 35 38 


2 50 6 


2 


43 2 


2 


16 43 


1 


38 17 


25 


2 


4 26 


2 36 26 


2 50 13 


2 


42 26 


2 


15 34 


1 


36 54 


26 


2 


5 43 


2 37 13 


2 50 19 


2 


41 49 


2 


14 25 


1 


35 32 


27 


2 


6 59 


2 37 59 


2 50 23 


2 


41 11 


2 


13 16 


1 


34 9 


23 


2 


8 15 


2 38 44 


2 50 25 


2 


40 31 


2 


12 5 


1 


32 46 


29 


2 


9 30 


2 39 28 


2 50 26 


2 


39 50 


2 


10 54 


1 


31 23 


30 


2 


10 43 


2 40 10 


2 50 25 


2 


39 8 


2 


9 42 


1 


30 



TABLE XXV. 

Moon's Equatorial Parallax. Argument. 



Arg. of the Evection. 





Os 


Is 


Hs 


Ills 


IVs 


Vs 


I 
1 


0° 


V 28" 


1' 


23" 


1' 9" 


0' 50" 


0' 32" 


0' 18" 


30° 


2 


1 28 


1 


22 


1 8 


49 


30 


18 


28 


4 


1 28 


1 


22 


1 7 


47 


29 


17 


26 


6 


1 28 


1 


21 


1 5 


46 


28 


17 


24 


8 


1 28 


1 


20 


1 4 


45 


27 


16 


22 


10 


1 28 


1 


19 


1 3 


44 


26 


16 


20 


12 


1 27 


1 


18 


1 2 


42 


25 


15 


18 


14 


1 27 


1 


17 


1 


41 


24 


15 


16 


16 


1 27 


1 


16 


59 


40 


24 


15 


14 


18 


1 26 


1 


15 


58 


39 • 


23 


14 


12 


20 


1 26 


1 


14 


57 


37 


22 


14 


10 


22 


1 25 


1 


13 


55 


36 


21 


14 


8 


24 


1 25 


1 


12 


54 


35 


20 


14 


6 


26 


1 24 


1 


11 


53 


34 


20 


14 


4 


28 


1 24 


1 


10 


51 


33 


19 


13 


2 


30 


1 23 


1 


9 


50 


32 


18 


13 





XIs 


Xs 


IXs 


VTIIs 


VIIs 


Vis 





TABLE XXIV. 

Evection. Argument. — Evection Corrected. 



37 





Vis 


VIIs 


VIIIs 




IXs i 

1 




Xs 


XIs 


Oo 


ic 


30' 0" 


0o 50' 18" 


0o 20' 52" 


0° 


9' 34" i 


0o 19' 50" 


0O49'16" 


1 


1 


28 37 





49 6 





20 10 





9 34 





20 32 





50 30 


2 


1 


27 14 





47 55 





19 29 





9 35 





21 16 





51 45 


3 


1 


25 51 





46 44 





18 49 





9 37 





22 1 





53 1 


4 


1 


24 28 





45 34 





18 11 





9 41 





22 47 





54 17 


5 


1 


23 6 





44 26 





17 34 





9 47 





23 34 





55 33 


6 


1 


21 43 





43 17 





16 58 





9 54 





24 22 





56 51 




1 


20 20 





42 10 





16 24 





10 2 





25 12 





58 9 


8 


1 


18 58 





41 4 





15 50 





10 12 





26 3 





59 28 


9 


1 


17 36 





39 58 





15 19 





10 23 





26 55 


1 


47 


10 


1 


16 14 





38 53 





14 48 





10 36 





27 48 


1 


2 7 


11 


1 


14 52 





37 49 





14 19 





10 50 





28 43 


1 


3 27 


12 


1 


13 31 





36 46 





13 51 





11 5 





29 39 


1 


4 48 


13 


1 


12 10 





35 44 





13 25 





11 23 


o 


30 35 


1 


6 9 


14 


1 


10 49 





34 43 





13 





11 41 





31 33 


1 


7 31 


15 


1 


9 29 





33 43 





12 37 





12 1 





32 32 


1 


8 53 


16 


1 


8 09 





32 44 





12 14 





12 23 





33 32 


1 


10 16 


17 


1 


6 49 





31 46 





11 54 





12 45 





34 34 


1 


11 39 


18 


1 


5 30 





30 49 





11 34 





13 10 





35 36 


1 


13 2 


19 


1 


4 11 





29 53 





11 16 





13 35 





36 39 


1 


14 26 


20 


1 


2 52 





28 58 





11 





14 3 





37 43 


1 


15 50 


21 


1 


1 34 





28 5 





10 45 





14 31 


o 


38 48 


1 


17 14 


22 


1 


17 





27 12 





10 31 





15 1 


o 


39 55 


1 


18 39 


23 





59 





26 20 





10 19 





15 33 





41 2 


1 


20 3 


24 


o 


57 44 





25 30 





10 8 





16 5 





42 10 


1 


21 28 


25 


o 


56 28 





24 40 





9 59 





16 39 


o 


43 19 


1 


22 53 


26 





55 13 





23 52 





9 51 





17 15 


o 


44 29 


1 


24 18 


27 





53 58 





23 5 





9 45 





17 52 





45 39 


1 


25 44 


28 





52 44 





22 20 





9 40 





18 30 





46 51 


1 


27 9 


29 





51 31 





21 35 





9 36 





19 9 





48 3 


1 


28 34 


30 





50 18 


1 


20 52 





9 34 





19 50 





49 16 


1 


30 



TABLE P. 

Moon's Equatorial Parallax. Argument. — Arg. of the Variation. 



~lp~ 


0s 


Is 


lis 


nis 


rvs 


Vs 




56" 


42" 


16" 


4" 


18" 


44" 


30o 


2 


55 


41 


14 


4 


19 


46 


28 


4 


55 


39 


13 


4 


21 


47 


26 


6 


55 


37 


12 


4 


23 


48 


24 


8 


55 


35 


10 


5 


24 


50 


22 


10 


54 


34 


9 


6 


26 


51 


20 


12 


53 


32 


8 


6 


28 


52 


18 


14 


52 


30 


7 


7 


30 


53 


16 


16 


51 


28 


6 


8 


32 


54 


14 


18 


50 


26 


6 


9 


34 


55 


12 


20 


49 


2-4 


5 


10 


35 


55 


10 


22 


48 


23 


4 


12 


37 


56 


8 


24 


47 


21 


4 


13 


39 


56 


6 


26 


45 


19 


4 


14 


41 


57 


4 


28 


44 


18 


4 


16 


42 


57 


2 


30 


42 


16 


4 


18 


44 


57 







XIs 


Xs 


IXs 


VIIIs 


VHs 


Vis 





38 TABLE XXV. 

Equation of Moon's Center. Argument. 



-Anomaly corrected. 





0s 


» 


lis 


Ills 


IVs 


Vs 


0° 


7c 


0' 0" 


10° 20' 58" 


12° 38' 44" 


13© 17' 35" 


12° 16' 21" 


9o 58 29" 


1 


7 


7 5 


10 


26 52 


12 


41 43 


13 


17 5 


12 


12 48 


9 


52 58 


! 2 


7 


14 10 


10 


32 42 


12 


44 35 


13 


16 28 


12 


9 11 


9 


47 24 


3 


7 


21 15 


10 


38 27 


12 


47 20 


13 


15 44 


12 


5 29 


9 


41 48 


4 


7 


28 19 


10 


44 8 


12 


49 59 


13 


14 53 


12 


1 41 


9 


36 10 


5 


7 


35 23 


10 


49 43 


12 


52 30 


13 


13 56 


11 


57 49 


9 


30 29 


6 


7 


42 26 


10 


55 14 


12 


54 55 


13 


12 52 


11 


53 52 


9 


24 46 


7 


7 


49 28 


11 


39 


12 


57 12 


13 


11 41 


11 


49 50 


9 


19 1 


8 


7 


56 28 


11 


6 


12 


59 23 


13 


10 24 


11 


45 44 


9 


13 13 


9 


8 


3 28 


11 


11 15 


13 


1 26 


13 


9 1 


11 


41 33 


9 


7 24 


10 


8 


10 26 


11 


16 24 


13 


3 23 


13 


7 31 


11 


37 17 


9 


1 32 


11 


8 


17 22 


11 


21 29 


13 


5 12 


13 


5 54 


11 


32 57 


8 


55 39 


12 


8 


24 17 


11 


26 27 


13 


6 55 


13 


4 12 


11 


28 33 


8 


49 44 


13 


8 


31 10 


11 


31 20 


13 


8 30 


13 


2 23 


11 


24 5 


8 


43 47 


14 


8 


38 1 


11 


36 8 


13 


9 59 


13 


27 


11 


19 32 


8 


37 49 


15 


8 


44 50 


11 


40 49 


13 


11 20 


12 


58 26 


1] 


14 55 


8 


31 49 


16 


8 


51 36 


11 


45 25 


13 


12 34 


12 


56 18 


11 


10 14 


8 


25 48 j 


17 


8 


58 20 


11 


49 54 


13 


13 41 


12 


54 5 


11 


5 30 


8 


19 46 


18 


9 


5 1 


11 


54 18 


13 


14 41 


12 


51 45 


11 


41 


8 


13 42 


19 


9 


11 39 


11 


58 35 


13 


15 34 


12 


49 19 


10 


55 49 


8 


7 38 


20 


9 


18 15 


12 


2 47 


13 


16 20 


12 


46 47 


10 


50 53 


8 


1 32 


21 


9 


24 47 


12 


6 52 


13 


16 59 


12 


44 10 


10 


45 53 


7 


55 26 


22 


9 


31 16 


12 


10 50 


13 


17 31 


12 


41 27 


10 


40 50 


7 


49 18 


23 


9 


37 42 


12 


14 42 


13 


17 56 


12 


38 38 


10 


35 43 


7 


43 10 


24 


9 


44 4 


12 


18 28 


13 


18 14 


12 


35 43 


10 


30 33 


7 


37 1 


25 


9 


50 23 


12 


22 7 


13 


18 24 


12 


32 43 


10 


25 20 


7 


30 52 


26 


9 


56 38 


12 


25 40 


13 


18 28 


12 


29 37 


10 


20 4 


7 


24 42 


27 


10 


2 49 


12 


29 6 


13 


18 25 


12 


26 26 


10 


14 45 


7 


18 32 


28 


10 


8 56 


12 


32 25 


13 


18 16 


12 


23 10 


10 


9 22 


7 


12 21 


29 


10 


14 59 


12 


35 38 


13 


17 59 


12 


19 48 


10 


3 57 


7 


6 11 


30 


10 


20 58 


12 


38 44 


13 


17 35 


12 


16 21 


9 


58 29 


7 






TABLE XXVI. 

Moon's Equatorial Parallax. Argument.- 



-Corrected Anomaly. 







0s 


Is 


lis 


Ills 


IVs 


Vs 




0° 


58' 


58''' 


58' 27" 


57' 


8" 


55' 


30" 


54' 


2" 


53' 


3" 


30° 


2 


58 


58 


58 23 


57 


2 


55 


23 


53 


57 


53 





28 


4 


58 


57 


58 19 


56 


55 


55 


17 


53 


52 


52 


58 


26 


6 


58 


56 


58 14 


56 


49 


55 


11 


53 


47 


52 


56 


24 


8 


58 


55 


58 10 


56 


42 


55 


4 


53 


43 


52 


54 


22 


10 


58 


54 


58 5 


56 


36 


54 


58 


53 


38 


52 


52 


20 


12 


58 


53 


58 


56 


29 


54 


52 


53 


34 


52 


50 


18 


14 


58 


51 


57 55 


56 


22 


54 


46 


53 


30 


52 


49 


16 


16 


58 


49 


57 49 


56 


16 


54 


40 


53 


26 


52 


47 


14 


18 


58 


46 


57 44 


56 


9 


54 


34 


53 


22 


52 


46 


12 


20 


58 


44 


57 38 


56 


3 


54 


29 


53 


19 


52 


45 


10 


22 


58 


41 


57 32 


55 


56 


54 


23 


53 


15 


52 


44 


8 


24 


58 


38 


57 26 


55 


49 


54 


18 


53 


12 


52 


43 


6 


26 


58 


34 


57 20 


55 


43 


54 


12 


53 


9 


52 


43 


4 


28 


58 


31 


57 14 


55 


36 


54 


7 


53 


6 


52 


43 


2 


30 


58 


27 


57 8 


55 


30 


54 2 
VIHs 


53 


3 


52 


43 





L— . -. — 


XIs 


Xs 


IXs 


VIIs 


1 





TABLE XXV I 

Equation of Moon's Center. Argument. — Anomaly corrected. 





Vis 


VIIs 


VIIIs 


IXs | 


Xs 


XIs 


0° 


70 


0' 0" 


40 


1' 31" 


ic 


43' 39 ' 


0o 42' 25" 


l c 


21' 16" 


3c 


39' 2" 


1 


6 


53 49 


3 


56 3 


1 


40 12 





42 1 


1 


24 22 


3 


45 1 


2 


6 


47 39 


3 


50 38 


1 


36 50 





41 44 


1 


27 35 


3 


51 4 


3 


6 


41 28 


3 


45 15 


1 


33 34 





41 35 


1 


30 54 


3 


57 11 


4 


6 


35 18 


3 


39 56 


1 


30 23 





41 32 


1 


34 20 


4 


3 22 


5 


6 


29 8 


3 


34 40 


1 


27 17 





41 36 


1 


37 53 


4 


9 37 


6 


6 


22 59 


3 


29 26 


1 


24 17 





41 46 


1 


41 32 


4 


15 55 




6 


16 50 


3 


24 17 


1 


21 22 





42 4 


1 


45 18 


4 


22 18 


8 


6 


10 42 


3 


19 10 


1 


18 33 





42 29 


1 


49 10 


4 


28 44 


9 


6 


4 34 


3 


14 7 


1 


15 50 





43 1 


1 


53 8 


4 


35 13 


10 


5 


58 28 


3 


9 7 


1 


13 12 





43 40 


1 


57 13 


4 


41 45 


11 


5 


52 22 


3 


4 11 


1 


10 41 





44 26 


2 


1 24 


4 


48 21 


12 


5 


46 17 


2 


59 19 


1 


8 15 





45 19 


2 


5 42 


4 


54 59 


13 


5 


40 14 


2 


54 30 


1 


5 55 





46 19 


2 


10 5 


5 


1 40 


14 


5 


34 12 


2 


49 46 


1 


3 42 





47 26 


2 


14 35 


5 


8 24 


15 


5 


28 11 


2 


45 5 


1 


1 34 





48 40 


2 


19 11 


5 


15 10 


16 


5 


22 11 


2 


40 28 





59 33 





50 1 


2 


23 52 


5 


21 59 i 


17 


5 


16 13 


2 


35 55 





57 37 





51 30 


2 


28 39 


5 


28 50 


18 


5 


10 16 


2 


31 27 





55 48 





53 5 


2 


33 32 


5 


35 43 


19 


5 


4 21 


2 


27 3 





54 6 





54 47 


2 


38 31 


5 


42 37 


20 


4 


58 28 


2 


22 43 





52 29 





56 37 


2 


43 35 


5 


49 34 


21 


4 


52 36 


2 


18 27 





50 59 





58 33 


2 


48 45 


5 


56 32 


22 


4 


46 47 


2 


14 16 





49 36 


1 


37 


2 


54 


6 


3 31 


23 


4 


40 59 


2 


10 10 





48 19 


1 


2 48 


2 


59 21 


6 


10 32 


24 


4 


36 14 


2 


6 8 





47 8 


1 


5 5 


3 


4 46 


6 


17 34 


25 


4 


29 31 


2 


2 11 





46 4 


1 


7 30 


3 


10 17 


6 


24 37 


26 


4 


23 50 


1 


58 19 





45 7 


1 


10 1 


3 


15 52 


6 


31 41 


27 


4 


18 11 


1 


54 31 





44 16 


1 


12 40 


3 


21 33 


6 


38 45 


28 


4 


12 35 


1 


50 49 





43 32 


1 


15 25 


3 


27 18 


6 


45 50 1 


29 


4 


7 2 


1 


47 11 





42 55 


1 


18 17 


3 


33 8 


6 


52 55 


30 


4 


1 31 


1 


43 39 





42 25 


1 


21 16 


3 


39 2 


7 






2c* 



40 



TABLE XXVII. 



VARIATION. 
Argument. — Variation, corrected. 





0s 


Is 


lis 


Ills 


IVs 


Vs 


o 


o ' 


a 


O ' 


a 


o 


/ 


it 


o 


/ 


a 


o 


/ 


a 


o 


/ II 





38 





1 8 


1 


1 


6 


58 





35 


54 





5 


29 





6 2 


2 


40 


26 


1 9 


7 


1 


5 


36 





33 


27 





4 


21 





7 24 


4 


42 


52 


1 10 


3 


1 


4 


5 





31 








3 


22 





8 55 


6 


45 


16 


1 10 


50 


1 


2 


27 





28 


34 





2 


33 





10 34 


8 


47 


38 


1 11 


26 


1 





42 





26 


11 





1 


54 





12 22 


10 


49 


57 


1 11 


53 





58 


49 





23 


51 





1 


24 





14 17 


12 


52 


13 


1 12 


9 





56 


50 





21 


34 





1 


5 





16 19 


14 


54 


24 


1 12 


15 





54 


45 





19 


22 








57 





18 27 


16 


56 


30 


1 12 


10 





52 


35 





17 


15 








59 





20 41 


18 


58 


30 


1 11 


55 





50 


21 





15 


13 





1 


11 





23 


20 


1 


24 


1 11 


30 





48 


2 





13 


17 





1 


34 





25 23 


22 


1 2 


11 


1 10 


55 





45 


40 





11 


28 





2 


8 





27 50 


24 


1 3 


51 


1 10 


10 





43 


16 





9 


47 





2 


51 





30 20 


26 


1 5 


23 


1 9 


15 





40 


50 





8 


13 





3 


45 





32 52 


28 


1 6 


47 


1 8 


11 





38 


22 





6 


47 





4 


48 





35 26 


30 


1 8 


1 


1 6 


58 





35 


54 





5 


26 





6 


2 





38 



■ 


Vis 


VIIs 


VIIIs 


IXs 


Xs 


XIs 


o 


o 


1 


a 


o 


/ 


a 


o 


/ 


a 


o ' 


a 


o 


/ 


a 


O ' 


a 








38 





1 


9 


58 


1 


10 


30 


40 


6 





9 


2 


7 


58 


2 





40 


34 


1 


11 


11 


1 


9 


13 


37 


38 





7 


49 


9 


13 


4 





43 


8 


1 


12 


15 


1 


7 


47 


35 


10 





6 


45 


10 


37 


6 





45 


40 


1 


13 


9 


1 


6 


13 


32 


44 





5 


50 


12 


9 


8 





48 


10 


1 


13 


52 


1 


4 


31 


30 


19 





5 


5 


13 


49 


10 





50 


37 


1 


14 


26 


1 


2 


42 


27 


58 





4 


29 


15 


36 


12 





53 





1 


14 


48 


1 





47 


25 


39 





4 


4 


17 


30 


14 





55 


19 


1 


15 


1 





58 


45 


23 


25 





3 


50 


19 


30 


16 





57 


33 


1 


15 


3 





56 


38 


21 


15 





3 


45 


21 


36 


18 





58 


41 


1 


14 


54 





54 


25 


19 


10 





3 


51 


23 


47 


20 


1 


1 


43 


1 


14 


35 





52 


9 


17 


11 





4 


7 


26 


3 


22 


1 


3 


38 


1 


14 


6 





49 


49 


15 


18 





4 


34 


28 


22 


24 


1 


5 


25 


1 


13 


27 





47 


26 


13 


33 





5 


10 


30 


44 


26 


1 


7 


5 


1 


12 


38 





45 





11 


54 





5 


57 


33 


8 


28 


1 


8 


36 


1 


11 


39 





42 


33 


10 


24 





6 


53 


35 


33 


30 


1 


9 


58 


1 


10 


30 





40 


6 


9 


2 





7 


58 


38 






TABLE XXVIII. 
moon's distance from the north pole of the ecliptic. 
Argument. Supplement of Node-j-Moon's Orbit Longitude. 



41 





nis 


IVs 


Vs 


Vis 


VIIs 


VIIIs 




0° 


84° S9' 16" 


85° 20' 


43" 


87° 


13' 


47" 


89° 


48' 0" 


92° 22' 13" 


94° 


15' 17" 


30° 


2 


84 


39 27 


85 


26 


16 


87 


23 


12 


89 


58 46 


92 


31 27 


94 


20 31 


28 


4 


84 


40 1 


85 


32 


9 


87 


32 


48 


90 


9 31 


92 


40 30 


94 


25 25 


26 


6 


84. 


40 58 


85 


38 


20 


87 


42 


33 


90 


20 14 


92 


49 19 


94 


29 59 


24 


8 


84 


42 17 


85 


44 


50 


88 


52 


28 


90 


30 55 


92 


57 56 


94 


34 12 


22 


10 


84 


43 58 


85 


51 


37 


88 


2 


31 


90 


41 33 


93 


6 18 


94 


38 4 


20 


12 


84 


46 2 


85 


58 


42 


88 


12 


42 


90 


52 7 


93 


14 27 


94 


41 35 


18 


14 


84 


48 27 


86 


6 


3 


88 


23 





91 


2 36 


93 


22 20 


94 


44 45 


16 


16 


84 


51 15 


86 


13 


40 


88 


33 


24 


91 


13 


93 


29 57 


94 


47 32 


14 


18 


84 


54 25 


86 


21 


33 


88 


43 


53 


91 


23 18 


93 


37 18 


94 


49 58 


12 


20 


84 


57 56 


86 


29 


42 


88 


54 


27 


91 


33 29 


93 


44 23 


94 


52 2 


10 


22 


85 


1 48 


86 


38 


4 


89 


5 


5 


91 


43 32 


93 


51 10 


94 


53 43 


8 


24 


88 


6 1 


86 


46 


41 


89 


15 


46 


91 


53 27 


93 


57 40 


94 


55 2 


6 


m 


85 


10 35 


86 


55 


30 


89 


26 


29 


92 


3 12 


94 


3 51 


94 


55 59 


4 


28 


85 


15 29 


87 


4 


32 


89 


37 


14 


92 


12 48 


94 


9 44 


94 


56 33 


2 


30 


85 


20 48 


87 


13 


47 


89 


48 





92 


22 13 


94 


15 17 


94 


56 44 







ns 


Is 


0s 


XIs 


Xs 




IXs 


i 



TABLE XXIX. 

EQUATION II OF THE MOON'S POLAR DISTANCE. 
Argument II, corrected. 





His 


IVs 


Vs 


Vis 


VHs 


VIHs 




0° 


0' 14" 


1' 24" 


4' 37" 


9' 0" 


13' 23" 


16' 36" 


30° 


2 


14 


1 34 


4 53 


9 18 


13 39 


16 45 


28 


4 


15 


1 44 


5 9 


9 37 


13 54 


16 53 


26 


6 


17 


1 54 


5 26 


9 55 


14 9 


17 1 


24 


8 


19 


2 5 


5 43 


10 13 


14 24 


17 S 


22 


10 


22 


2 17 


6 


10 31 


14 38 


17 14 


20 


12 


25 


2 29 


6 17 


10 49 


14 52 


17 20 


18 


14 


29 


2 41 


6 35 


11 7 


15 5 


17 26 


16 


16 


34 


2 54 


6 53 


11 25 


15 18 


17 31 


14 


18 


40 


3 8 


7 11 


11 43 


15 31 


17 35 


12 


20 


45 


3 22 


7 29 


12 


15 43 


17 38 


10 


22 


52 


3 36 


7 47 


12 17 


15 55 


17 41 


8 


24 


59 


3 51 


8 5 


12 34 


16 6 


17 43 


6 


26 


1 7 


4 6 


8 23 


12 51 


16 16 


17 45 


4 


28 


1 15 


4 21 


8 42 


13 7 


16 26 


17 46 


2 


30 


1 24 


4 37 


9 


13 23 


16 36 


17 46 







ns 


Is 


Os 


XIs 


Xs 


IXs 





TABLE XXX. 

EQUATION III OF THE POLAR DISTANCE. 
Argument. Moon's True Longitude. 





His 


IVs 


Vs 


Vis 


VIIs 


VTHs 




0° 
6 
12 

18 
24 
30 


16" 

16 

16 

16 
15 
15 


15" 

14 

14 

13 
13 
12 


12" 
11 

10 

10 
9 

8 


8" 

7 

6 

5 
5 
4 


4" 

3 

3 

2 
1 
1 


1" 

1 








30° 
24 

18 

12 

6 



■ 


ns 


Is 


Os 


XIs 


Xs 


IXs 





23 



42 TABLE XXXI. 

EQUATIONS OF POLAR DISTANCE. 
Arguments. — 20 of Longitude; V to IX, corrected; and X, not corrected. 



Arg. 


20 


V. 


VI. 


VII. 


VIII. 


IX. 


X 


Arg. 


260 


0" 


56" 


6" 


3" 


25" 


3" 


11" 


240 


280 


1 


55 


6 


3 


25 


3 


11 


220 


300 


1 


55 


7 


4 


25 


4 


11 


200 


320 


2 


53 


8 


5 


24 


6 


32 


180 


340 


3 


52 


10 


6 


23 


7 


13 


160 


360 


4 


50 


12 


8 


23 


9 


14 


140 


380 


5 


48 


14 


10 


22 


11 


16 


120 


400 


6 


45 


16 


12 


21 


14 


17 


100 


420 


8 


42 


18 


14 


20 


17 


19 


80 


440 


10 


39 


21 


17 


19 


20 


21 


60 ; 


460 


11 


36 


24 


19 


17 


23 


23 


40 


480 


13 


33 


27 


22 


16 


27 


25 


20 


500 


15 


30 


30 


25 


15 


30 


27 


000 


520 


17 


27 


33 


28 


14 


33 


29 


980 


540 


19 


24 


36 


31 


12 


37 


31 


960 


560 


20 


20 


39 


33 


11 


40 


33 


940 


580 


22 


17 


41 


36 


10 


43 


35 


920 


600 


24 


15 


44 


38 


9 


46 


37 


900 


620 


25 


12 


46 


40 


8 


48 


38 


880 


640 


26 


10 


48 


42 


7 


51 


40 


860 


660 


27 


8 


50 


44 


6 


53 


41 


840 


680 


28 


7 


52 


45 


6 


54 


42 


820 


700 


29 


5 


53 


46 


5 


56 


42 


800 


720 


29 


5 


53 


47 


5 


56 


43 


780 


740 


30 


4 


54 


47 


5 


57 


43 


760 



TABLE XXXII. 

REDUCTION 
Argument. — Supplement of Node -J- Moon's Orbit Longitude. 





0s 


Vis 


Is VIIs 


lis 


VIIIs 


Ills 


IXs 


IVt 


Xs 


Vs 


XIs 


0° 


7' 


0" 


1' 3" 


1' 


3" 


7' 


0" 


13' 


57' 


12' 


57" 


2 


6 


31 


49 


1 


18 


7 


29 


13 


10 


12 


42 


4 


6 


3 


38 


1 


35 


7 


57 


13 


22 


12 


25 


6 


5 


34 


28 


1 


54 


8 


26 


13 


32 


12 


6 


8 


5 


6 


20 


2 


14 


8 


54 


13 


40 


11 


46 


10 


4 


39 


14 


2 


35 


9 


21 


13 


46 


11 


25 


12 


4 


12 


10 


2 


58 


9 


48 


13 


50 


11 


2 


14 


3 


46 


8 


3 


22 


10 


13 


13 


52 


10 


38 


16 


3 


22 


8 


3 


46 


10 


38 


13 


52 


10 


13 


18 


2 


58 


10 


4 


]2 


11 


2 


13 


50 


9 


48 


20 


2 


35 


14 


4 


39 


11 


25 


13 


46 


9 


21 


22 


2 


14 


20 


5 


6 


11 


46 


13 


40 


8 


54 


24 


1 


54 


28 


5 


34 


12 


6 


13 


32 


8 


26 


26 


1 


35 


38 


6 


3 


12 


25 


13 


22 


7 


57 


28 


1 


18 


49 


6 


31 


12 


42 


13 


10 


7 


29 


30 


1 


3 


1 3 


7 





12 


57 1 


12 


57 


7 


, 



TABLE XXXIV. 

moon's semidiameter. 
Argument. Equatorial Parallax. 



43 



Eq. Parallax. 


53' 


0" 


53 


20 


53 


40 


54 





54 


20 


54 


40 


55 





55 


20 


55 


40 


56 






Semidiam. 



14' 27" 

14 32 

14 37 

14 43 

14 48 

14 54 

14 59 

15 5 
15 10 
15 16 



Eq. Parallax. 



56' 0" 

56 20 

56 40 

57 
57 20 

57 40 

58 
58 20 

58 40 

59 



Semidiam. 



15' 16" 

15 21 

15 26 

15 32 

15 37 

15 43 

15 48 

15 54 

15 59 

16 5 



Eq. Parallax 



59' 0" 

59 20 

59 40 

60 
60 20 

60 40 

61 
61 20 

61 40 

62 



Semidiam. 


16' 


5" 


16 


10 


16 


16 


16 


21 


16 


26 


16 


32 


16 


37 


16 


43 


16 


48 


16 


54 



TABLE XXXV. 

AUGMENTATION OF MOON'S SEMI- 
DIAMETER. 
Argument. Apparent Altitude. 



Ap. Alt. 


Angm. 


6° 


2" 


12 


3 


18 


5 


24 


6 


30 


8 


36 


9 


42 


11 


48 


12 


54 


13 


60 


14 


66 


15 


72 


15 


78 


16 


84 


16 


90 


16 







TABLE XXXVI. 

moon's hourly motion in lon- 
gitude. 

Arguments. 2, 3, 4, and 5 of Lon- 
gitude. 



Arg. 


2 


3 


4 


5 


Arg. 





6" 


1" 


3' 


3 


100 


5 


5 


2 


3 


3 


95 


10 


5 


2 


3 


3 


90 


15 


4 


2 


3 


3 


85 


20 


4 


3 


2 


2 


80 


25 


3 


3 


2 


2 


75 


30 


2 


3 


2 


2 


70 


35 


2 


4 


1 


1 


65 


40 


1 


4 


1 


1 


60 


45 


1 


4 


1 


1 


55 


50 





5 


1 


1 


50 



TABLE XXXVII. 

moon's hourly motion in longitude 
Argument. Argument of the Evection. 



'■ 


0s 


Is 


Us 


IHs 


rvs 


Vs 




0° 


V 20" 


1' 15" 


1' 0" 


0' 39" 


0' 20" 


0' 6" 


30° 


2 


1 20 


1 14 


58 


38 


19 


5 


28 


4 


1 20 


1 13 


57 


37 


18 


5 


26 


6 


1 20 


1 12 


56 


35 


16 


4 


24 


8 


1 20 


1 11 


54 


34 


15 


4 


22 


10 


1 20 


1 11 


53 


33 


14 


3 


20 


12 


1 19 


. 10 


52 


31 


13 


3 


18 


14 


1 19 


1 9 


50 


30 


12 


2 


16 


16 


1 19 


1 8 


49 


29 


11 


2 


14 


18 


1 18 


1 7 


48 


27 


11 


2 


12 


20 


1 18 


1 5 


46 


26 


10 


1 


10 


22 


1 17 


1 4 


45 


25 


9 


1 


8 


24 


1 17 


1 3 


44 


23 


8 


1 


6 


26 


1 16 


1 2 


42 


22 


7 


1 


4 


28 


1 15 


1 1 


41 


21 


7 


1 


2 


30 


1 15 


1 


39 


20 


6 


1 







Xls 


Xs 


IXs 


VHIs 


VIIs 


Vis 



44 TABLE XXXVIII. 

moon's hourly motion in longitude. 
Arguments. Sum of preceding equations, and Anomaly, corrected. 



j 


0" 


20" 


40" 


60" 


80" 


100" 


1 


Os 


0° 


4" 


6" 


9" 


11" 


14" 


16" 


XHs 


0° 




10 


4 


7 


9 


11 


13 


16 




20 




20 


5 


7 


9 


11 


13 


15 




10 


Is 





5 


7 


9 


11 


13 


15 


XIs 







10 


6 


7 


9 


11 


13 


14 




20 




20 


7 


8 


9 


11 


12 


13 




10 


lis 





7 


8 


9 


11 


12 


13 


Xs 







10 


8 


9 


10 


10 


11 


12 




20 




20 


9 


10 


10 


10 


10 


11 




10 


THs 





10 


10 


10 


10 


10 


10 


IXs 







10 


11 


11 


10 


10 


9 


9 




20 




20 


12 


11 


10 


10 


9 


8 




10 


IVs 





13 


12 


11 


9 


8 


7 


vnis 







10 


14 


12 


11 


9 


8 


6 




20 




20 


14 


12 


11 


9 


8 


6 




10 


Vs 





15 


13 


11 


9 


7 


5 


VHs 







10 


15 


13 


11 


9 


7 


5 




20 




20 


15 


13 


11 


9 


7 


5 




10 


Vis 





15 


13 


11 


9 


7 


5 


Vis 







0" 


20" 


40" 


60" 


80" 


100" 





TABLE XXXIX. 

moon's hourly motion in longitude. 

Argument. Anomaly, corrected. 





0s 


Is 


Hs 


IHs 


IVs 


Vs 




0° 


34' 51" 


34' 14" 


32' 39" 


30' 45" 


29' 6" 


28' 1" 


30° 


2 


34 51 


34 9 


32 32 


30 38 


29 


27 58 


28 


4 


34 51 


34 4 


32 24 


30 31 


28 55 


27 55 


26 


6 


34 50 


33 59 


32 17 


30 23 


28 50 


27 53 


24 


8 


34 49 


33 53 


32 9 


30 16 


28 45 


27 50 


22 


10 


34 47 


33 47 


32 2 


30 9 


28 40 


27 48 


20 


12 


34 45 


33 41 


31 54 


30 2 


28 35 


27 46 


18 


14 


34 43 


33 35 


31 46 


29 56 


28 30 


27 45 


16 


16 


34 41 


33 28 


31 38 


29 49 


28 26 


27 43 


14 


18 


34 38 


33 22 


31 31 


29 42 


28 22 


27 42 


12 


20 


34 34 


33 15 


31 23 


29 36 


28 18 


27 41 


10 


22 


34 31 


33 8 


31 15 


29 30 


28 14 


27 40 


8 


24 


34 27 


33 1 


31 8 


29 23 


28 10 


27 39 


6 


26 


34 23 


32 54 


31 


29 17 


28 7 


27 39 


4 


28 


34 19 


32 47 


30 53 


29 12 


28 4 


27 38 


2 


30 


34 14 


32 39 


30 45 


29 6 


28 1 


27 38 







XIs 


Xs 


IXs 


vnis 


VHs 


Vis 





TABLE XL. 
moon's hourly motion in longitude. 

Arguments. Sum of preceding equations, and Argument of Variation. 



45 





27' 


29' 


31' 


33' 


35' 


37' 




Os 


0° 


0" 


2" 


5" 


7" 


10" 


12" 


XTTs 


0° 




10 





3 


5 


7 


9 


12 




20 




20 


1 


3 


5 


7 


9 


11 




10 


Is 





3 


4 


5 


7 


8 


9 


XIs 







10 


5 


5 


6 


6 


7 


7 




20 




20 


7 


7 


6 


6 


5 


5 




10 


Hs 





9 


8 


7 


5 


4 


3 


Xs 







10 


11 


9 


7 


5 


3 


1 




20 




20 


12 


10 


7 


5 


2 







10 


ms 





12 


10 


7 


5 


2 





IXs 







10 


12 


10 


7 


5 


2 







20 




20 


11 


9 


7 


5 


3 


1 




10 


IVs 





9 


8 


7 


5 


4 


3 


vms 







10 


7 


7 


6 


6 


5 


5 




20 




20 


5 


5 


6 


6 


7 


7 




10 


Vs 





3 


4 


5 


7 


8 


9 


VHs 







10 


1 


3 


5 


7 


9 


11 




20 




20 





2 


5 


7 


10 


12 




10 


Vis 








2 


5 


7 


10 


12 


Vis 







27' 


29' 


31' 


33' 


35' 


37' 


..., 



TABLE XLI. 

MOON'S HOURLY MOTION IN LONGITUDE. 

Argument. Argument of the Variation. 





0s 


Is 


Hs 


IHs 


IVs 


Vs 




3 


1' 17" 


0' 58" 


0' 20" 


0' 2" 


0' 22" 


1' 0" 


30° 


2 


1 17 


55 


18 


3 


24 


1 2 


28 


4 


1 17 


53 


16 


3 


26 


1 4 


26 


6 


1 16 


51 


14 


3 


29 


1 6 


24 


8 


1 16 


48 


12 


4 


31 


1 8 


22 


10 


1 15 


45 


11 


5 


34 


1 10 


20 


12 


1 14 


43 


9 


6 


37 


1 12 


18 


14 


1 13 


40 


8 


7 


39 


1 13 


16 


16 


1 11 


38 


6 


8 


42 


1 15 


14 


18 


1 10 


35 


5 


10 


44 


1 16 


12 


20 


1 8 


32 


4 


11 


47 


1 17 


10 


22 


1 6 


30 


4 


13 


50 


1 18 


8 


24 


1 4 


27 


3 


15 


52 


1 18 


6 


26 


1 2 


25 


3 


17 


55 


1 19 


4 


28 


1 


23 


2 


19 


57 


1 19 


2 


30 


58 


20 


2 


22 


1 


1 19 







XIs 


Xs 


IXs 


VIHs 


VHs 


Vis 





TABLE XLII. 



MOON'S HOURLY MOTION IN LONGITUD 
Argument. Argument of the Reduction. 





0s 


Is 


lis 


Ills 


IVs 


Vs 




0° 


2" 


6" 


14" 


18" 


14" 


6" 


30° 


2 


2 


7 


14 


18 


13 


6 


28 


4 


2 


7 


15 


18 


13 


5 


26 


6 


2 


8 


15 


18 


12 


5 


24 


8 


2 


8 


16 


18 


12 


4 


22 


10 


3 


9 


16 


17 


11 


4 


20 


12 


3 


9 


16 


17 


11 


4 


18 


14 


3 


10 


17 


17 


10 


3 


16 


16 


3 


10 


17 


17 


10 


3 


14 


18 


4 


11 


17 


16 


9 


3 


12 


20 


4 


11 


17 


16 


9 


3 


10 


22 


4 


12 


18 


16 


8 


2 


8 


24 


5 


12 


18 


15 


8 


2 


6 


26 


5 


13 


18 


15 


7 


2 


4 


28 


6 


13 


18 


14 


7 


2 


2 


30 


6 


14 


18 


14 


6 


2 







XIs 


Xs 


IXs 


VIIIs 


VIIs 


Vis 





TABLE XLIII. 

moon's hourly motion in latitude. 
Argument. Argument I, of Latitude. 



[ 


0s+ 


Is+ 


IIs+ 


IIIs— 


IVs— 


Vs— 




0° 


2' 58" 


2' 34" 


V 29" 


0' 0" 


V 29" 


2' 34" 


30° 


2 


2 58 


2 31 


1 24 


6 


1 35 


2 37 


28 


4 


2 58 


2 28 


1 18 


12 


1 40 


2 40 


26 


6 


2 57 


2 24 


1 13 


19 


1 45 


2 43 


24 


8 


2 56 


2 20 


1 7 


25 


1 50 


2 45 


22 


10 


2 55 


2 17 


1 1 


31 


1 55 


2 47 


20 


12 


2 54 


2 12 


55 


37 


1 59 


2 49 


18 


14 


2 53 


2 8 


49 


43 


2 4 


2 51 


16 


16 


2 51 


2 4 


43 


49 


2 8 


2 53 


14 


18 


2 49 


1 59 


37 


55 


2 12 


2 54 


12 


20 


2 47 


1 55 


31 


1 1 


2 17 


2 55 


10 


22 


2 45 


1 50 


25 


1 7 


2 20 


2 56 


8 


24 


2 43 


1 45 


19 


1 13 


2 24 


2 57 


6 


26 


2 40 


1 40 


12 


1 18 


2 28 


2 58 


4 


28 


2 37 


1 35 


6 


1 24 


2 31 


2 58 


2 


30 


2 34 


1 29 





1 29 


2 34 


2 58 







XIs+ 


Xs-f 


IXs-f 


VIIIs— 


VIIs— 


Vis— 





TABLE XLIV. 

moon's hourly motion in latitude. 

Argument. Argument II, of Latitude. 





0s+ 


Is+ 


IIs+ 


IIIs— 


IVs— 


Vs— 




0° 


4" 


4" 


2" 


0" 


2" 


4" 


30° 


6 


4 


3 


2 





3 


4 


24 


12 


4 


3 


1 


1 


3 


4 


18 


18 


4 


9 


1 


1 


3 


4 


12 


24 


4 


3 





2 


3 


4 


6 


30 


4 


2 





2 


4 


4 







XIs+ 


Xs+ 


IXs+ 


VHIs— 


VHs— 


Vis— 


1 



TABLE XLV.— PROPORTIONAL LOGARITHMS. 47 





i 
0' 


1/ 


2' 


3' 


4' 


1 
5' 


I 
6' 


7' 


0" 


ooooo 


~ 17782 


14771 


13010 


11761 


10792 


10000 


9331 


1 


35563 


17710 


14735 


12986 


11743 


10777 


9988 


9320 


2 


32553 


17639 


14699 


12962 


11725 


10763 


9976 


9310 


3 


3 0792 


17570 


14664 


12939 


11707 


10749 


9964 


9300 


4 


29542 


17501 


14629 


12915 


11689 


10734 


9952 


9289 


5 


28573 


17434 


14594 


12891 


11671 


10720 


9940 


9273 


6 


27782 


17368 


14559 


12868 


11654 


10706 


992S 


9269 


7 


27112 


17302 


14525 


12845 


11636 


10b92 


9916 


9259 


8 


26532 


17238 


14491 


12821 


11619 


10678 


9905 


9249 


9 


26021 


17175 


14457 


12798 


11601 


10663 


9S93 


9238 


10 


25563 


17112 


14424 


12775 


11584 


10649 


9881 


9228 


11 


25149 


17050 


14390 


12753 


11566 


10635 


9869 


9218 


12 


24771 


16990 


14357 


12730 


11549 


10621 


9858 


9208 


13 


24424 


16930 


14325 


12707 


11532 


10608 


9846 


9198 


14 


24102 


16871 


14292 


12685 


11515 


10594 


9834 


9188 


15 


23802 


16812 


14260 


12663 


11498 


10580 


9828 


9178 


16 


23522 


16755 


14228 


12640 


11481 


10566 


9811 


9168 


17 


23259 


16698 


14196 


12618 


11464 


10552 


9800 


9158 


18 


23010 


16642 


14165 


12596 


11457 


10539 


9788 


9148 


19 


22775 


16587 


14133 


12574 


11430 


10525 


9777 


9138 


20 


22553 


16532 


14102 


12553 


11413 


10512 


9765 


9128 


21 


22341 


16478 


14071 


12531 


11397 


10498 


9754 


9119 


22 


22139 


16425 


14040 


12510 


11380 


10484 


9742 


9109 


23 


21946 


16372 


14010 


12488 


11363 


10471 


9731 


9099 


24 


21761 


16320 


13979 


12467 


11347 


10458 


9720 


9089 


25 


21584 


16269 


13949 


12445 


11331 


10444 


9708 


9079 


26 


21413 


16218 


13919 


12424 


11314 


10431 


9697 


9070 


27 


21249 


16168 


13890 


12403 


11298 


10418 


9686 


9060 


28 


21091 


16118 


13860 


123S2 


11282 


10404 


9675 


9050 


29 


20939 


16069 


13831 


12362 


11266 


10391 


9664 


9041 


30 


20792 


16021 


13802 


12341 


11249 


10378 


9652 


9031 


31 


20649 


15973 


13773 


12320 


1 1233 


10365 


9641 


9021 


32 


20512 


15925 


13745 


12300 


11217 


10352 


9630 


9012 


33 


20378 


15878 


13716 


12279 


11201 


10339 


9619 


9002 


34 


20248 


15832 


13688 


12259 


11186 


10326 


9608 


8992 


35 


20122 


15786 


13660 


12239 


11170 


10313 


9597 


8983 


36 


20000 


15740 


13632 


12218 


11154 


10300 


9586 


8973 


37 


19881 


15695 


13604 


12198 


11138 


10287 


9575 


8964 


38 


19765 


15651 


13576 


12178 


11123 


10274 


9564 


8954 


39 


19652 


15607 


13549 


12159 


11107 


10261 


9553 


8945 


40 


19542 


15563 


13522 


12139 


11091 


10248 


9542 


8935 


41 


19435 


15520 


13495 


12119 


11076 


10235 


9532 


8926 


42 


19331 


15477 


13468 


12099 


11061 


10223 


9521 


8917 


43 


19228 


15435 


13441 


12080 


1 1045 


10210 


9510 


8907 


44 


19128 


15393 


13415 


12061 


11030 


10197 


9499 


8898 


45 


19031 


15351 


13388 


12041 


11015 


10185 


9488 


8888 


46 


18935 


15310 


13362 


12022 


10999 


10172 


9478 


8879 


47 


18842 


15269 


13336 


12003 


10984 


10160 


9467 


8870 


48 


18751 


15229 


13310 


1 1984 


10969 


10147 


9456 


8861 


49 


18661 


15189 


13284 


11965 


10954 


10135 


9446 


8851 


50 


18573 


15149 


13259 


11946 


10939 


10122 


9435 


8842 


51 


18487 


15110 


13233 


11927 


10924 


10110 


9425 


8833 


52 


18403 


15071 


13208 


11908 


10909 


10098 


9414 


8824 


53 


1 8320 


15032 


13183 


11889 


10894 


10085 


9404 


8814 


54 


18239 


14994 


13158 


11871 


10880 


10073 


9393 


8805 


55 


1,8? 59 


14956 


13133 


11852 


10865 


10061 


9383 


8796 


56 


18081 


14918 


13108 


11834 


10850 


10049 


9372 


8787 


57 


18004 


14881 


13083 


11816 


10835 


10036 


9362 


8778 


58 


17929 


14844 


13059 


11797 


1 0821 


10024 


9351 


8769 


59 


17855 


14808 


13034 


11779 


10806 


10012 


9341 


8760 


60 


17782 


14771 


13010 

1 


11761 


10792 

1 


10000 


9331 


8751 



2d 



TABLE XL V.— PROPORTIONAL LOGARITHMS. 





8' 


0" 


8751 


1 


8742 


2 


8733 


3 


8724 


4 


8715 


5 


8706 


6 


8697 


7 


8688 


8 


8679 


9 


8670 


10 


8661 


11 


8652 


12 


8643 


13 


8635 


14 


8626 


15 


8617 


16 


8608 


17 


8599 


18 


8591 


19 


8582 


20 


8573 


21 


8565 


22 


8556 


23 


8547 


24 


8539 


25 


8530 


26 


8522 


27 


8513 


28 


8504 


29 


8496 


30 


8487 


31 


8479 


32 


8470 


33 


8462 


34 


8453 


35 


8445 


36 


8437 


37 


8128 


38 


8420 


39 


8411 


40 


8403 


41 


8395 


42 


8386 


43 


8378 


44 


8370 


45 


8361 


46 


8353 


47 


8345 


48 


8337 


49 


8328 


50 


8320 


51 


8312 


52 


8304 


59 


8296 


54 


8288 


55 


8279 


56 


8271 


57 


8263 


68 


8255 


59 


8247 


6C 


8239 



8239 
8231 
8223 
8215 

8207 
8199 

8191 
8183 
8175 
8167 
8159 

8152 
8144 
8136 
8128 
8120 

8112 
8104 

8097 



8073 
8066 
8058 
8050 
8043 

8035 
8027 
8020 
8012 



7997 
7989 
7981 
7974 
7966 

7959 
7951 
7944 
7936 
7929 

7921 
7914 
7906 
7899 
7891 

7884 
7877 
7869 
7862 
7855 

7847 
7840 
7832 
7825 
7818 

7811 

7803 
7796 
7789 
7782 



10' 


11' 


12' 


13' 


14' 


7782 


7368 


6990 


6642 


6320 


7774 


7361 


6984 


6637 


6315 


7767 


7354 


6978 


6631 


6310 


7760 


7348 


6972 


6625 


6305 


7753 


7341 


6966 


6620 


6300 


7745 


7335 


6960 


6614 


&d94 


7738 


7328 


6954 


6609 


6289 


7731 


7322 


6948 


6603 


6284 


7724 


7315 


6942 


6598 


6279 


7717 


7309 


6936 


6592 


6274 


7710 


7302 


6930 


6587 


6269 


7703 


7296 


6924 


6581 


6264 


7696 


7289 


6918 


6576 


6259 


7688 


7283 


6912 


6570 


6254 


7681 


7276 


6906 


6565 


6248 


7674 


7270 


6900 


6559 


6243 


7667 


7264 


6894 


6554 


6238 


7660 


7257 


6888 


6548 


6233 


7653 


7251 


6882 


6543 


6228 


7646 


7244 


6877 


6538 


6223 


7639 


7238 


6871 


6532 


6218 


7632 


7232 


6865 


6527 


6213 


7625 


7225 


6859 


6521 


6208 


7618 


7219 


6853 


6516 


6203 


7611 


7212 


6847 


6510 


6198 


7604 


7206 


6841 


6505 


6193 


7597 


7200 


6836 


6500 


6188 


7590 


7193 


6830 


6494 


6183 


7583 


7187 


6824 


6489 


6178 


7577 


7181 


6818 


6484 


6173 


7570 


7175 


6812 


6478 


6168 


7563 


7168 


6807 


6473 


6163 


7556 


7162 


6801 


6467 


6158 


7549 


7156 


6795 


6462 


6153 


7542 


7149 


6789 


6457 


6148 


7535 


7143 


6784 


6451 


6143 


7528 


7137 


6778 


6446 


6138 


7522 


7131 


6772 


6441 


6133 


7515 


7124 


6766 


6435 


6128 


7508 


7118 


6761 


6430 


6123 


7501 


7112 


6755 


6425 


6118 


7494 


7106 


6749 


6420 


6113 


7488 


7100 


6743 


6414 


6108 


7481 


7093 


6738 


6409 


6103 


7474 


7087 


6732 


6404 


6099 


7467 


7081 


6726 


6398 


6094 


7461 


7075 


6721 


6393 


6089 


7454 


7069 


6715 


6388 


6084 


7447 


7063 


6709 


6383 


6079 


7441 


7057 


6704 


6377 


6074 


7434 


7050 


6698 


6372 


6069 


7427 


7044 


6692 


6367 


6064 


7421 


7038 


6687 


6362 


6059 


7414 


7032 


6681 


6357 


6055 


7407 


7026 


6676 


6351 


6050 


7401 


7020 


6670 


6346 


6045 


7394 


7014 


6664 


6341 


6040 


7387 


7008 


6659 


6336 


6035 


7381 


7002 


6653 


6331 


6030 


7374 


6996 


6648 


6325 


6025 


7368 


6990 


6642 


6320 


6021 



15' 



6021 
6016 
6011 
6006 
6001 
5997 

5992 
5987 
5982 
5977 
5973 

5968 
5963 
5958 
5954 
5949 

5944 
5939 
5935 
5930 
5925 

5920 
5916 
5911 
5906 
5902 

5897 
5892 
5888 
5883 
5878 

5874 
5869 
5864 
5860 
5855 

5850 
5846 
5841 
5836 
5832 

5827 
5823 
5818 
5813 
5809 

5804 
5800 
5795 
5790 
5786 

5781 
5777 
5772 
5768 
5763 

5758 
5754 
5749 
5745 
5740 



16' 



5740 
5736 
5731 
5727 
5722 
5718 

5713 
5709 
5704 
5700 
5695 

5691 
5686 
5682 
5677 
5673 

5669 
5664 
5660 
5655 
5651 

5646 
5642 
5637 
5633 
5629 

5624 
5620 
5615 
5611 
5607 

5602 
5598 
5594 
5589 
5585 

5580 
5576 
5572 
5567 
5563 

5559 
5554 
5550 
5546 
5541 

5537 
5533 
5528 
5524 
5520 

5516 
5511 
5507 
5503 
5498 

5494 
5490 

5486 
5481 

5477 



TABLE XLV.— PROPORTIONAL LOGARITHMS, 



49 



l 


17' 


18' 


19' 


20' 


21' 


22' 


23' 


24' 


25' 


0" 


5477 


"5229 


4994 


4771 


4559 


4357 


4164 


3979 


3802 


1 


5473 


5225 


4990 


4768 


4556 


4354 


4161 


3976 


3799 


2 


5469 


5221 


4986 


4764 


4552 


4351 


415.8 


3973 


3796 


3 


5464 


5217 


4983 


4760 


4549 


4347 


4155 


3970 


3793 


4 


5460 


5213 


4979 


4757 


4546 


4344 


4152 


3967 


3791 


5 


5456 


5209 


4975 


4753 


4542 


4341 


4] 49 


3964 


3788 


6 


5452 


5205 


4971 


4750 


4539 


4338 


4145 


3961 


3785 


7 


5447 


5201 


4967 


4746 


4535 


4334 


4142 


3958 


3782 


8 


5443 


5197 


4964 


4742 


4532 


4331 


4139 


3955 


3779 


9 


5439 


5193 


4960 


4739 


4528 


4328 


4136 


3952 


3776 


10 


5435 


5189 


4956 


4735 


4525 


4325 


4133 


3949 


3773 


11 


5430 


5185 


4952 


4732 


4522 


4321 


4130 


3946 


3770 


12 


5426 


5181 


4949 


4728 


4518 


4318 


4127 


3943 


3768 


13 


5422 


5177 


4945 


4724 


4515 


4315 


4124 


3940 


3765 


14 


5418 


5173 


4941 


4721 


4511 


4311 


4120 


3937 


3762 


15 


5414 


5169 


4937 


4717 


4508 


4308 


4117 


8934 


3759 


IS 


5409 


5165 


4933 


4714 


4505 


4305 


4114 


3931 


3758 


17 


5405 


5161 


4930 


4710 


4501 


4302 


4111 


3928 


3753 


18 


5401 


5157 


4926 


4707 


4498 


4298 


4108 


3925 


3750 


19 


5397 


5153 


4922 


4703 


4494 


4295 


4105 


3922 


3747 


20 


5393 


5149 


4918 


4699 


4491 


4292 


4102 


3919 


3745 


21 


5389 


5145 


4915 


4696 


4488 


4289 


4099 


3917 


3742 


22 


5384 


5141 


4911 


4692 


4484 


4285 


4096 


3914 


3739 


23 


5380 


5137 


4907 


4689 


4481 


4282 


4092 


3911 


3736 


24 


5376 


5133 


4903 


4685 


4477 


4279 


4089 


3908 


3733 


25 


5372 


5129 


4900 


4682 


4474 


4276 


4086 


3905 


3730 


26 


5368 


5125 


4896 


4678 


4471 


4273 


4083 


3902 


3727 


27 


5364 


5122 


4892 


4675 


4467 


4269 


4080 


3899 


3725 


28 


5359 


5118 


4889 


4671 


4464 


4266 


4077 


3896 


3722 


29 


5355 


5114 


4885 


4668 


4460 


4263 


4074 


3893 


3719 


30 


5351 


5110 


4881 


4664 


4457 


4260 


4071 


3890 


3716 


31 


5347 


5106 


4877 


4660 


4454 


4256 


4068 


3887 


3713 


32 


5343 


5102 


4874 


4657 


4450 


4253 


4065 


3884 


3710 


33 


5339 


5098 


4870 


4653 


4447 


4250 


4062 


3881 


3708 


34 


5335 


5C94 


4866 


4650 


4444 


4247 


4059 


3878 


3705 


35 


5331 


5090 


4863 


4646 


4440 


4244 


4055 


3875 


3702 


36 


5326 


5086 


4859 


4643 


4437 


4240 


4052 


3872 


3699 


37 


5322 


5082 


4855 


4639 


4434 


4237 


4049 


3869 


3696 


38 


5318 


5079 


4852 


■ 4636 


4430 


4234 


4046 


3866 


3693 


39 


5314 


5075 


4848 


4632 


4427 


4231 


4043 


3863 


3691 


40 


5310 


5071 


4844 


4629 


4424 


4228 


4040 


3860 


3688 


41 


5306 


5067 


4841 


4625 


4420 


4224 


4037 


3857 


3685 


42 


5302 


5063 


4837 


4622 


4417 


4221 


4034 


3855 


3682 


43 


5298 


5059 


4833 


4618 


4414 


4218 


4031 


3852 


3679 


44 


5294 


5055 


4830 


4615 


4410 


4215 


4028 


3S49 


3677 


45 


5290 


5051 


4828 


4611 


4407 


4212 


4025 


3846 


3674 


46 


5285 


5048 


4822 


4608 


4404 


4209 


4022 


3843 


3671 


47 


5281 


5044 


4819 


4604 


4400 


4205 


4019 


3840 


3668 


48 


5277 


5040 


4815 


4601 


4397 


4202 


4016 


3837 


3665 


49 


5273 


5036 


4811 


4597 


4394 


4199 


4013 


3834 


3663 


50 


5269 


5032 


4808 


4594 


4390 


4196 


4010 


3831 


3660 


51 


5265 


5028 


4804 


4590 


4387 


4193 


4007 


3828 


3657 


52 


5261 


5025 


4800 


4587 


4384 


4189 


4004 


3825 


3654 


53 


5257 


5021 


4797 


4584 


4380 


4186 


4001 


3922 


3651 


54 


5253 


5017 


4793 


4580 


4377 


4183 


3998 


3820 


3649 


55 


5249 


5013 


4789 


4577 


4374 


4180 


3995 


3317 


3646 


50 


5245 


o009 


4786 


4573 


4370 


4177 


3991 


3814 


3643 


57 


5241 


5005 


4782 


4570 


4367 


4174 


3988 


3811 


3640 


58 


5237 


5002 


4778 


4566 


4364 


4171 


3985 


3808 


3637 


59 


5233 


4998 


4775 


4563 


4361 


4167 


3982 


3805 


3635 


60 


5229 


4994 


4771 


4559 


4357 


4164 


3979 


4802 


3632 



50 TABLE XLV.— PROPORTIONAL LOGARITHMS, 



! 


26' 


27' 


28' 


29' 


30' 


31' 


32' 


33' 


34' 


0" 


3632 


3468 


3310 


3158 


3010 


2868 


2730 


2596 


2467 


1 


3629 


3465 


3307 


3155 


3008 


2866 


2728 


2594 


2465 


2 


3626 


3463 


3305 


3153 


3005 


2863 


2725 


2592 


2462 


3 


3623 


3460 


3302 


3150 


3003 


2861 


2723 


2590 


2460 


4 


3621 


3457 


3300 


3148 


3001 


2859 


2721 


2588 


2458 


5 


3618 


3454 


3297 


3145 


2998 


2856 


2719 


2585 


2456 


6 


3615 


3452 


3294 


3143 


2996 


2854 


2716 


2583 


2454 


7 


3612 


3449 


3292 


3140 


2993 


2852 


2714 


2581 


2452 


8 


3610 


3446 


3289 


3138 


2991 


2849 


2712 


2579 


2450 


9 


3607 


3444 


3287 


3135 


2989 


2-847 


2710 


2577 


2448 


10 


3604 


3441 


3284 


3133 


2986 


2845 


2707 


2574 


2445 


11 


3601 


3438 


3282 


3130 


2984 


2842 


2705 


2572 


2443 


12 


3598 


3436 


3279 


3128 


2981 


2840 


2703 


2570 


2441 


13 


3596 


3433 


3276 


3125 


2979 


2838 


2701 


2568 


2439 


14 


3593 


3431 


3274 


3123 


2977 


2835 


2698 


2566 


2437 


1 15 


3590 


3428 


3271 


3120 


2974 


2833 


2696 


2564 


2435 


! 16 


3587 


3425 


3269 


3118 


2972 


2831 


2694 


2561 


2433 


17 


3585 


3423 


3266 


3115 


2969 


2828 


2692 


2559 


2431 


18 


3582 


3420 


3264 


3113 


2967 


2826 


2689 


2557 


2429 


19 


3579 


3417 


3261 


3110 


2965 


2824 


2687 


2555 


2426 


20 


3576 


3415 


3259 


3108 


2962 


2821 


2685 


2553 


2424 


21 


3574 


3412 


3256 


3105 


2960 


2819 


2683 


2451 


2422 


22 


3571 


3409 


3253 


3103 


2958 


2817 


2681 


2548 


2420 


23 


3568 


3407 


3251 


3101 


2955 


2815 


2678 


2546 


2418 


24 


3565 


3404 


3248 


3098 


2953 


2812 


2676 


2544 


2416 


25 


3563 


3401 


3246 


3096 


2950 


2810 


2674 


2542 


2414 


26 


3560 


3399 


3243 


3093 


2948 


2808 


2672 


2540 


2412 


27 


3557 


3396 


3241 


3091 


2946 


2805 


2669 


2538 


2410 


2S 


3555 


3393 


3238 


3088 


2943 


2803 


2667 


2535 


2408 


29 


3552 


3391 


3236 


3086 


2941 


2801 


2665 


2533 


2405 


30 


3549 


3388 


3233 


3083 


2939 


2798 


2663 


2531 


2403 


31 


3546 


3386 


3231 


3081 


2936 


2796 


2060 


2529 


2401 


32 


3544 


3383 


3228 


3078 


2934 


2794 


2658 


2527 


2399 


33 


3541 


3380 


3225 


3076 


2931 


2792 


2656 


2525 


2397 


34 


3538 


3378 


3223 


3073 


2929 


2789 


2654 


2522 


2395 


35 


3535 


3375 


3220 


3071 


2927 


2787 


2652 


2520 


2393 


36 


3533 


3372 


3218 


3069 


2924 


2785 


2649 


2518 


2391 


37 


3530 


3370 


3215 


3066 


2922 


2782 


2647 


2516 


2389 


38 


3527 


3367 


3213 


3064 


2920 


2780 


2645 


2514 


2387 


39 


3525 


3365 


3210 


3061 


2917 


2778 


2643 


2512 


2384 


40 


3522 


3362 


3208 


3059 


2915 


2775 


2640 


2510 


2382 


41 


3519 


3359 


3205 


3056 


2912 


2773 


2638 


2507 


2380 


42 


3516 


3357 


3203 


3054 


2910 


2771 


2636 


2505 


2378 


43 


3514 


3354 


3200 


3052 


2908 


2769 


2634 


2503 


2376 


44 


3511 


3351 


3198 


3049 


2905 


2766 


2632 


2501 


2374 


45 


3508 


3349 


3195 


3047 


2903 


2764 


2629 


2499 


2372 


46 


3506 


3346 


3193 


3044 


2901 


2762 


2627 


2497 


2370 


47 


3503 


3344 


3190 


3042 


2898 


2760 


2625 


2494 


2368 


48 


3500 


3341 


3188 


3039 


2896 


2757 


2623 


2492 


2366 


49 


3497 


3338 


3185 


3037 


2894 


2755 


2621 


2490 


2364 


50 


3495 


3336 


3183 


3034 


2891 


2753 


2618 


2488 


2362 


51 


3492 


3333 


3180 


3032 


2889 


2750 


2616 


2486 


2359 


52 


3489 


3331 


3178 


3030 


2887 


2748 


2614 


2484 


2357 


53 


3487 


3328 


3175 


3027 


2884 


2746 


2812 


2482 


2355 


54 


3484 


3325 


3173 


3025 


2882 


2744 


2610 


2480 


2353 


55 


3481 


3323 


3170 


3022 


2880 


2741 


2607 


2477 


2351 


56 


3479 


3320 


3168 


3020 


2877 


2739 


2605 


2475 


2349 


57 


3476 


3318 


3165 


3018 


2875 


2737 


2603 


2473 


2347 


58 


3473 


3315 


3163 


3015 


2873 


2735 


2601 


2471 


2345 


59 


3471 


3313 


3160 


3013 


2870 


2732 


2599 


2469 


2343 


60 


3468 


3310 


3158 | 


3010 


2868 


2730 


2596 


2467 


2341 



TABLE XL V.— PROPORTIONAL LOGARITHMS. 51 





35' ! 


26' 

i 


37' 


28' 


39' 


40' 


41' 


42' 


43' 


1 

a" 


2341 


2218 


2099 


1984 


1871 


1761 


1654 


1549 


1447 


1 


2339 


2216 


2098 


19S2 


1869 


1759 


1652 


1547 


1445 


2 ; 


2337 , 


2214 


2096 


1980 


1867 


1757 


1650 


1546 


1443 


3 


2:335 


2212 


2094 


1978 


1865 


1755 


1648 


1544 


1442 


■ 4 


2233 1 


2210 


2092 


1976 


1863 


1754 


1647 


1542 


1440 


5 


2331 


2208 


2090 


1974 


1862 


1752 


1645 


1540 


1438 


6 


2328 ! 


2206 


2088 


1972 


1860 


1750 


1643 


1539 


1437 


7 


2326 


2204 


2086 


1970 


1858 


1748 


1641 


1537 i 


14.35 


8 


2324 


2202 


2084 


1968 


1856 


1746 


1640 


1535 


1433 


9 


2322 


2200 


2032 


1967 


1854 


1745 


1638 


1534 


1432 


10 


2320 


2198 


2080 


1965 


1852 


1743 


1636 


1532 


1430 


11 


2318 


2196 


2078 


1963 


1850 


1741 


1634 


1530 


1428 


12 


2316 


2194 


2076 


1961 


1849 


1739 


1633 


1528 


1427 


13 


2314 


2192 


2074 


1959 


1847 


1737 


1631 


1527 


1425 


14 


2312 


2190 


2072 


1957 


1845 


1736 


1629 


1525 


1423 


15 


2310 


2188 


2070 


1955 


1843 


1734 


1627 


1523 


1422 


16 


2308 


2186 


2068 


1953 


1841 


1732 


1626 


1522 


1420 


17 


2306 


2184 


2066 


1951 


1839 


1730 


1624 


1520 


1418 


18 


2304 


2182 


2064 


1950 


1838 


1723 


1622 


1518 


1417 


19 


2302 


2180 


2062 


1948 


1836 


1727 


1620 


1516 


1415 


20 


2300 


2178 


2061 


1946 


1834 


1725 


1619 


1515 


1413 


21 


2298 


2176 


2059 


1944 


1832 


1723 


"1617 


1513 


1412 


22 


2296 


2174 


2057 


1942 


1830 


1721 


1615 


1511 


1410 


23 


2294 


2172 


2055 


1940 


1828 


1719 


1613 


1510 


1408 


24 


2291 


2170 


2053 


1938 


1827 


1718 


1612 


1508 


1407 


25 


2289 


2169 


2051 


1936 


1825 


1716 


1610 


1506 


1405 


26 


2287 


2167 


2049 


1934 


1823 


1714 


1608 


1504 


T403 


27 


2285 


2165 


2047 


1933 


1821 


1712 


1606 


1503 


1402 


28 


2283 


2163 


2045 


1931 


1819 


1711 


1605 


1501 


1400 


29 


2281 


2161 


2043 


1929 


1817 


1709 


1603 


1499 


1398 


30 


2279 


2159 


2041 


1927 


1816 


1707 


1601 


1498 


1397 


31 


2277 


2157 


2039 


1925 


1814 


1705 


1599 


1496 


1395 


32 


2275 


2155 


2037 


1923 


1812 


1703 


1598 


1494 


1393 


33 


2273 


2153 


2035 


1921 


1810 


1702 


1596 


1493 


1392 


34 


2271 


2151 


2033 


1919 


1808 


1700 


1594 


1491 


1390 


35 


2269 


2149 


2032 


1918 


1806 


1698 


1592 


1489 


1388 


36 


2267 


2147 


2030 


1916 


1805 


1696 


1591 


1487 


1387 


37 


2265 


2145 


2028 


1914 


1803 


1694 


1589 


1486 


1385 


38 


2263 


2143 


2026 


1912 


1801 


1893 


1587 


1484 


1383 


39 


2261 


2141 


2024 


1910 


1799 


1691 


1585 


1482 


1382 


40 


2259 


2139 


2022 


1908 


1797 


1689 


1584 


1481 


1380 


41 


2257 


2137 


2020 


1906 


1795 


1687 


1582 


1479 


1378 


42 


2255 


2135 


201S 


1904 


1794 


1686 


1580 


1477 


1377 


43 


2253 


2133 


2016 


1903 


1792 


1684 


1578 


1476 


1375 


44 


2251 


2131 


2014 


1901 


1790 


1682 


1577 


1474 


1373 


45 


2249 


2129 


2012 


1899 


1788 


1680 


1575 


1472 


1372 


46 


2247 


2127 


2010 


1897 


1786 


1678 


1573 


1470 


1370 


47 


2245 


2125 


2039 


1895 


1785 


1677 


1571 


1469 


1368 


48 


2243 


2123 


2007 


1893 


1783 


1675 


1570 


1467 


1367 


49 


2241 


2121 


2005 


1891 


1781 


1673 


1568 


1465 


1365 


50 


2239 


2119 


2003 


1889 


1779 


1671 


1566 


1464 


1363 


51 


2237 


2117 


2001 


.888 


1777 


1670 


1565 


1462 


1362 


52 


2235 


2115 


1999 


1886 


1775 


1668 


1563 


1460 


1360 


53 


2233 


2113 


1997 


1884 


1774 


1666 


1561 


1459 


1359 


54 


2231 


2111 


1995 


1882 


1772 


1664 


1559 


1457 


1357 


55 


2229 


2109 


1993 


1880 


1770 


1663 


1558 


1455 


1355 


56 


2227 


2107 


1991 


1878 


1768 


1661 


1556 


1454 


1354 


57 


2225 


2J05 


1989 


1876 


1766 


1659 


1554 


1452 


1352 


58 


2223 


2103 


1987 


1875 


1765 


1657 


1552 


1450 


1350 


59 


2220 


2101 


1986 


1873 


1763 


1655 


1551 


1449 


1349 


60 


2218 


2TO9 


1984 


1871 


1761 


1654 


1549 


1447 


1347 



23 



2d* 



0'2 TABLE 


XLV. 


—PROPORTIONAL LOGARITHMS. 




44' 


45' 


46' 


47' 


48' 


49' 


50' 


51' 


52' 


0" 


1347 


1249 


1154 


1061 


969 


880 


792 


706 


621 


i 


1345 


1248 


1152 


1059 


968 


878 


790 


704 


620 


2 


1344 


1246 


1151 


1057 


966 


877 


789 


703 


619 


3 


1342 


1245 


1149 


1056 


965 


875 


787 


702 


617 


4 


1340 


1243 


1148 


1054 


963 


874 


786 


700 


616 


5 


1339 


1241 


1146 


1053 


962 


872 


785 


699 


615 


6 


1337 


1240 


1145 


1051 


960 


871 


783 


697 


613 


7 


1335 


1238 


1143 


1050 


959 


769 


782 


696 


612 


8 


1334 


1237 


1141 


1048 


957 


868 


780 


694 


610 


9 


1332 


1235 


1140 


1047 


956 


866 


779 


693 


609 


10 


1331 


1233 


1138 


1045 


954 


865 


777 


692 


608 


11 


1329 


1232 


1137 


1044 


953 


"863 


776 


690 


606 


12 


1327 


1230 


1135 


1042 


951 


862 


774 


689 


605 


13 


1326 


1229 


1134 


1041 


950 


860 


773 


687 


603 


14 


1324 


1227 


1132 


1039 


948 


859 


772 


686 


602 


15 


1322 


1225 


1130 


1037 


947 


857 


770 


685 


601 


18 


1321 


1224 


1129 


1036 


945 


856 


769 


683 


599 


17 


1319 


1222 


1127 


1034 


944 


855 


767 


682 


598 


18 


1317 


1221 


1126 


1033 


942 


853 


766 


680 


596 


19 


1316 


1219 


1124 


1031 


941 


852 


764 


679 


595 


20 


1314 


1217 


1123 


1030 


939 


850 


763 


678 


594 


21 


1313 


1216 


1121 


1028 


938 


849 


762 


676 


592 


22 


1311 


1214 


1119 


1027 


936 


S47 


760 


675 


591 


23 


1309 


1213 


1118 


1025 


935 


846 


759 


673 


590 


24 


1308 


1211 


1116 


1024 


933 


844 


757 


672 


588 


25 


1306 


1209 


1115 


1022 


932 


843 


756 


670 


587 


26 


1304 


1208 


1113 


1021 


930 


841 


754 


669 


585 


27 


1303 


1206 


1112 


1019 


929 


840 


753 


668 


584 


28 


1301 


1205 


1110 


1018 


927 


838 


751 


666 


583 


29 


1300 


1203 


1109 


1016 


926 


837 


750 


665 


581 


30 


1298 


1201 


1107 


1015 


924 


835 


749 


663 


580 


31 


1296 


1200 


1105 


1013 


923 


834 


747 


662 


579 


32 


1295 


1198 


1104 


1012 


921 


833 


746 


661 


577 


33 


1293 


1197 


1102 


1010 


920 


831 


744 


659 


576 


34 


1291 


1195 


1101 


1008 


918 


830 


743 


658 


574 


35 


1290 


1193 


1099 


1007 


917 


828 


741 


656 


573 


36 


1288 


1192 


1098 


1005 


915 


827 


740 


655 


572 


37 


1287 


1190 


1096 


1004 


914 


825 


739 


654 


570 


38 


1285 


1189 


1095 


1002 


912 


824 


737 


652 


569 


39 


1283 


1187 


1093 


1001 


911 


822 


736 


651 


568 


40 


1282 


1186 


1091 


999 


909 


821 


734 


649 


566 


41 


1280 


1184 


1090 


998 


908 


819 


733 


648 


565 


42 


1278 


1182 


1088 


996 


906 


818 


731 


647 


563 


43 


1277 


1181 


1087 


995 


905 


816 


730 


645 


562 


44 


1275 


1179 


1085 


993 


903 


815 


729 


644 


561 


45 


1274 


1178 


1084 


992 


902 


814 


727 


642 


559 


46 


1272 


1176 


1082 


990 


900 


812 


726 


641 


558 


47 


1270 


1174 


1081 


989 


899 


811 


724 


640 


557 


48 


1269 


1173 


1079 


987 


897 


809 


723 


638 


555 


49 


1267 


1171 


1078 


986 


896 


808 


721 


637 


554 


50 


1266 


1170 


1076 


984 


894 


806 


720 


635 


552 


51 


1264 


1168 


1074 


983 


893 


805 


719 


634 


551 


52 


1262 


1167 


1073 


981 


891 


803 


717 


633 


550 


53 


1261 


1165 


1071 


980 


890 


802 


716 


631 


548 


54 


1259 


1163 


1070 


978 


888 


801 


714 


680 


547 


55 


1257 


1162 


1068 


977 


877 


799 


713 


628 


546 


36 


1256 


1160 


1067 


"975 


885 


798 


711 


627 


544 


57 


1254 


1159 


1065 


974 


884 


796 


710 


626 


553 


58 


1253 


1157 


1064 


972 


883 


795 


709 


624 


541 


59 


1251 


1156 


1062 


971 


881 


793 


707 


623 


540 


eo 


1249 


1154 


1061 I 969 


880 


792 


706 


621 


539 



TABLE XLV.— PROPORTIONAL LOGARITHMS. 





53' 


54' 


55' 


58' 


57' 


58' 


59' 


0" 


539 


438 


378 


300 


228 


147 


73 


1 


537 


456 


377 


298 


221 


146 


72 


2 


536 


455 


375 


297 


220 


145 


71 


3 


535 


454 


374 


296 


219 


148 


69 


4 


533 


452 


373 


294 


218 


142 


68 


5 


532 


451 


371 


293 


216 


141 


67 


6 


531 


450 


370 


292 


215 


140 


66 


7 


529 


448 


369 


291 


214 


139 


64 


8 


528 


447 


367 


289 


213 


137 


63 


9 


526 


446 


366 


288 


211 


136 


62 


10 


525 


444 


365 


287 


910 


135 


61 


11 


524 


443 


363 


285 


209 


134 


60 


12 


522 


442 


362 


284 


208 


132 


58 


13 


521 


440 


361 


283 


206 


131 


57 


14 


520 


439 


359 


282 


205 


130 


56 


15 


518 


438 


358 


280 


204 


129 


55 


*6 


517 


436 


357 


279 


302 


127 


53 


17 


516 


435 


356 


278 


201 


126 


52 


18 


514 


434 


354 


276 


200 


125 


51 


19 


513 


432 


353 


275 


199 


124 


50 


20 


512 


431 


352 


274 


197 


122 


49 


21 


510 


430 


350 


273 


106 


121 


47 


22 


509 


428 


349 


271 


195 


120 


46 


23 


507 


427 


348 


270 


194 


119 


45 


24 


506 


426 


346 


269 


192 


117 


44 


25 


505 


424 


345 


267 


191 


116 


42 


26 


503 


423 


344 


266 


190 


115 


41 


27 


502 


422 


342 


265 


189 


114 


40 


28 


501 


420 


341 


264 


187 


112 


39 


29 


499 


419 


340 


262 


186 


111 


38 


30 


498 


418 


339 


261 


185 


110 


36 


31 


497 


416 


337 


260 


184 


109 


35 


32 


495 


415 


336 


258 


182 


107 


34 


33 


494 


414 


335 


257 


181 


106 


33 


34 


493 


412 


333 


256 


180 


105 


31 


35 


491 


411 


332 


255 


179 


104 


30 


36 


490 


410 


331 


253 


177 


103 


29 


37 


489 


408 


329 


252 


176 


101 


28 


38 


487 


407 


328 


251 


175 


100 


27 


39 


486 


406 


327 


250 


174 


99 


25 


40 


484 


404 


326 


248 


172 


96 


24 


41 


483 


403 


324 


247 


171 


96 


23 


42 


482 


402 


323 


246 


170 


, 95 


22 


43 


480 


400 


322 


244 


169 


94 


21 


44 


479 


399 


320 


243 


167 


93 


19 


45 


478 


398 


319 


242 


166 


91 


18 


46 


476 


396 


318 


241 


165 


90 


17 


47 


475 


395 


316 


239 


163 


89 


16 


48 


474 


394 


315 


238 


162 


88 


15 


49 


472 


392 


314 


237 


161 


87 


13 


50 


471 


391 


313 


235 


160 


85 


12 


51 


470 


390 


311 


234 


158 


84 


11 


82 


468 


388 


310 


233 


157 


83 


10 


53 


467 


387 


309 


232 


156 


82 


8 


54 


466 


386 


307 


230 


155 


80 


7 


55 


464 


384 


308 


229 


153 


79 


6 


56 


463 


383 


305 


228 


152 


78 


5 


57 


462 


882 


304 


227 


151 


77 


4 


58 


460 


381 


302 


225 


150 


75 


2 


59 


459 


379 


801 


224 


148 


74 


1 


60 


458 


378 


300 


223 


147 


73 






spq. 




P^, 



^ 










> 



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